id	sid	tid	token	lemma	pos
ejpam-574	1	1	6_574_jafari.dvi	6_574_jafari.dvi	NUM
ejpam-574	1	2	european	european	ADJ
ejpam-574	1	3	journal	journal	NOUN
ejpam-574	1	4	of	of	ADP
ejpam-574	1	5	pure	pure	ADJ
ejpam-574	1	6	and	and	CCONJ
ejpam-574	1	7	applied	apply	VERB
ejpam-574	1	8	mathematics	mathematic	NOUN
ejpam-574	1	9	vol	vol	NOUN
ejpam-574	1	10	.	.	PROPN
ejpam-574	1	11	4	4	NUM
ejpam-574	1	12	,	,	PUNCT
ejpam-574	1	13	no	no	INTJ
ejpam-574	1	14	.	.	NOUN
ejpam-574	1	15	2	2	NUM
ejpam-574	1	16	,	,	PUNCT
ejpam-574	1	17	2011	2011	NUM
ejpam-574	1	18	,	,	PUNCT
ejpam-574	1	19	147	147	NUM
ejpam-574	1	20	-	-	SYM
ejpam-574	1	21	151	151	NUM
ejpam-574	1	22	issn	issn	PROPN
ejpam-574	1	23	1307	1307	NUM
ejpam-574	1	24	-	-	SYM
ejpam-574	1	25	5543	5543	NUM
ejpam-574	1	26	–	–	PUNCT
ejpam-574	1	27	www.ejpam.com	www.ejpam.com	X
ejpam-574	1	28	generalized	generalize	VERB
ejpam-574	1	29	closed	close	VERB
ejpam-574	1	30	sets	set	NOUN
ejpam-574	1	31	with	with	ADP
ejpam-574	1	32	respect	respect	NOUN
ejpam-574	1	33	to	to	ADP
ejpam-574	1	34	an	an	DET
ejpam-574	1	35	ideal	ideal	ADJ
ejpam-574	1	36	s.	s.	PROPN
ejpam-574	1	37	jafari1	jafari1	PROPN
ejpam-574	1	38	,	,	PUNCT
ejpam-574	1	39	n.	n.	PROPN
ejpam-574	1	40	rajesh2,∗	rajesh2,∗	ADP
ejpam-574	1	41	1	1	NUM
ejpam-574	1	42	college	college	NOUN
ejpam-574	1	43	of	of	ADP
ejpam-574	1	44	vestsjaelland	vestsjaelland	PROPN
ejpam-574	1	45	south	south	NOUN
ejpam-574	1	46	,	,	PUNCT
ejpam-574	1	47	herrestraede	herrestraede	NOUN
ejpam-574	1	48	11	11	NUM
ejpam-574	1	49	,	,	PUNCT
ejpam-574	1	50	4200	4200	NUM
ejpam-574	1	51	slagelse	slagelse	NOUN
ejpam-574	1	52	,	,	PUNCT
ejpam-574	1	53	denmark	denmark	NOUN
ejpam-574	1	54	.	.	PUNCT
ejpam-574	2	1	2	2	NUM
ejpam-574	2	2	rajah	rajah	NOUN
ejpam-574	2	3	serfoji	serfoji	NOUN
ejpam-574	2	4	govt	govt	NOUN
ejpam-574	2	5	.	.	PUNCT
ejpam-574	3	1	college	college	NOUN
ejpam-574	3	2	,	,	PUNCT
ejpam-574	3	3	thanjavur-613005	thanjavur-613005	NOUN
ejpam-574	3	4	,	,	PUNCT
ejpam-574	3	5	tamilnadu	tamilnadu	ADJ
ejpam-574	3	6	,	,	PUNCT
ejpam-574	3	7	india	india	PROPN
ejpam-574	3	8	.	.	PUNCT
ejpam-574	4	1	abstract	abstract	PROPN
ejpam-574	4	2	.	.	PUNCT
ejpam-574	5	1	an	an	DET
ejpam-574	5	2	ideal	ideal	NOUN
ejpam-574	5	3	on	on	ADP
ejpam-574	5	4	a	a	DET
ejpam-574	5	5	set	set	NOUN
ejpam-574	5	6	x	x	PUNCT
ejpam-574	5	7	is	be	AUX
ejpam-574	5	8	a	a	DET
ejpam-574	5	9	non	non	X
ejpam-574	5	10	empty	empty	ADJ
ejpam-574	5	11	collection	collection	NOUN
ejpam-574	5	12	of	of	ADP
ejpam-574	5	13	subsets	subset	NOUN
ejpam-574	5	14	of	of	ADP
ejpam-574	5	15	x	x	PUNCT
ejpam-574	5	16	with	with	ADP
ejpam-574	5	17	heredity	heredity	NOUN
ejpam-574	5	18	property	property	NOUN
ejpam-574	5	19	which	which	PRON
ejpam-574	5	20	is	be	AUX
ejpam-574	5	21	also	also	ADV
ejpam-574	5	22	closed	close	VERB
ejpam-574	5	23	under	under	ADP
ejpam-574	5	24	finite	finite	ADJ
ejpam-574	5	25	unions	union	NOUN
ejpam-574	5	26	.	.	PUNCT
ejpam-574	6	1	the	the	DET
ejpam-574	6	2	concept	concept	NOUN
ejpam-574	6	3	of	of	ADP
ejpam-574	6	4	generalized	generalized	ADJ
ejpam-574	6	5	closed	closed	ADJ
ejpam-574	6	6	sets	set	NOUN
ejpam-574	6	7	was	be	AUX
ejpam-574	6	8	introduced	introduce	VERB
ejpam-574	6	9	by	by	ADP
ejpam-574	6	10	levine	levine	PROPN
ejpam-574	6	11	.	.	PUNCT
ejpam-574	7	1	in	in	ADP
ejpam-574	7	2	this	this	DET
ejpam-574	7	3	paper	paper	NOUN
ejpam-574	7	4	,	,	PUNCT
ejpam-574	7	5	we	we	PRON
ejpam-574	7	6	introduce	introduce	VERB
ejpam-574	7	7	and	and	CCONJ
ejpam-574	7	8	investigate	investigate	VERB
ejpam-574	7	9	the	the	DET
ejpam-574	7	10	concept	concept	NOUN
ejpam-574	7	11	of	of	ADP
ejpam-574	7	12	generalized	generalized	ADJ
ejpam-574	7	13	closed	close	VERB
ejpam-574	7	14	sets	set	NOUN
ejpam-574	7	15	with	with	ADP
ejpam-574	7	16	respect	respect	NOUN
ejpam-574	7	17	to	to	ADP
ejpam-574	7	18	an	an	DET
ejpam-574	7	19	ideal	ideal	NOUN
ejpam-574	7	20	.	.	PUNCT
ejpam-574	8	1	2000	2000	NUM
ejpam-574	8	2	mathematics	mathematic	NOUN
ejpam-574	8	3	subject	subject	NOUN
ejpam-574	8	4	classifications	classification	NOUN
ejpam-574	8	5	:	:	PUNCT
ejpam-574	8	6	54c10	54c10	NUM
ejpam-574	8	7	key	key	ADJ
ejpam-574	8	8	words	word	NOUN
ejpam-574	8	9	and	and	CCONJ
ejpam-574	8	10	phrases	phrase	NOUN
ejpam-574	8	11	:	:	PUNCT
ejpam-574	8	12	topological	topological	ADJ
ejpam-574	8	13	spaces	space	NOUN
ejpam-574	8	14	,	,	PUNCT
ejpam-574	8	15	generalized	generalize	VERB
ejpam-574	8	16	closed	close	VERB
ejpam-574	8	17	set	set	NOUN
ejpam-574	8	18	,	,	PUNCT
ejpam-574	8	19	ideal	ideal	ADJ
ejpam-574	8	20	.	.	PUNCT
ejpam-574	9	1	1	1	X
ejpam-574	9	2	.	.	X
ejpam-574	9	3	introduction	introduction	NOUN
ejpam-574	9	4	indeed	indeed	ADV
ejpam-574	9	5	ideals	ideal	NOUN
ejpam-574	9	6	are	be	AUX
ejpam-574	9	7	very	very	ADV
ejpam-574	9	8	important	important	ADJ
ejpam-574	9	9	tools	tool	NOUN
ejpam-574	9	10	in	in	ADP
ejpam-574	9	11	general	general	ADJ
ejpam-574	9	12	topology	topology	NOUN
ejpam-574	9	13	.	.	PUNCT
ejpam-574	10	1	it	it	PRON
ejpam-574	10	2	was	be	AUX
ejpam-574	10	3	the	the	DET
ejpam-574	10	4	works	work	NOUN
ejpam-574	10	5	of	of	ADP
ejpam-574	10	6	newcomb	newcomb	PROPN
ejpam-574	11	1	[	[	X
ejpam-574	11	2	8	8	NUM
ejpam-574	11	3	]	]	PUNCT
ejpam-574	11	4	,	,	PUNCT
ejpam-574	11	5	rancin	rancin	VERB
ejpam-574	11	6	[	[	X
ejpam-574	11	7	9	9	NUM
ejpam-574	11	8	]	]	PUNCT
ejpam-574	11	9	,	,	PUNCT
ejpam-574	11	10	samuels	samuel	NOUN
ejpam-574	12	1	[	[	X
ejpam-574	12	2	10	10	NUM
ejpam-574	12	3	]	]	PUNCT
ejpam-574	12	4	and	and	CCONJ
ejpam-574	12	5	hamlet	hamlet	PROPN
ejpam-574	12	6	and	and	CCONJ
ejpam-574	12	7	jankovic	jankovic	PROPN
ejpam-574	12	8	(	(	PUNCT
ejpam-574	12	9	see	see	VERB
ejpam-574	12	10	[	[	X
ejpam-574	12	11	1	1	NUM
ejpam-574	12	12	,	,	PUNCT
ejpam-574	12	13	2	2	NUM
ejpam-574	12	14	,	,	PUNCT
ejpam-574	12	15	3	3	NUM
ejpam-574	12	16	,	,	PUNCT
ejpam-574	12	17	4	4	NUM
ejpam-574	12	18	,	,	PUNCT
ejpam-574	12	19	5	5	NUM
ejpam-574	12	20	]	]	PUNCT
ejpam-574	12	21	)	)	PUNCT
ejpam-574	12	22	which	which	PRON
ejpam-574	12	23	motivated	motivate	VERB
ejpam-574	12	24	the	the	DET
ejpam-574	12	25	research	research	NOUN
ejpam-574	12	26	in	in	ADP
ejpam-574	12	27	applying	apply	VERB
ejpam-574	12	28	topological	topological	ADJ
ejpam-574	12	29	ideals	ideal	NOUN
ejpam-574	12	30	to	to	PART
ejpam-574	12	31	generalize	generalize	VERB
ejpam-574	12	32	the	the	DET
ejpam-574	12	33	most	most	ADV
ejpam-574	12	34	basic	basic	ADJ
ejpam-574	12	35	properties	property	NOUN
ejpam-574	12	36	in	in	ADP
ejpam-574	12	37	general	general	ADJ
ejpam-574	12	38	topology	topology	NOUN
ejpam-574	12	39	.	.	PUNCT
ejpam-574	13	1	a	a	DET
ejpam-574	13	2	nonempty	nonempty	ADJ
ejpam-574	13	3	collection	collection	NOUN
ejpam-574	13	4	i	i	PRON
ejpam-574	13	5	of	of	ADP
ejpam-574	13	6	subsets	subset	NOUN
ejpam-574	13	7	on	on	ADP
ejpam-574	13	8	a	a	DET
ejpam-574	13	9	topological	topological	ADJ
ejpam-574	13	10	space	space	NOUN
ejpam-574	13	11	(	(	PUNCT
ejpam-574	13	12	x	x	X
ejpam-574	13	13	,	,	PUNCT
ejpam-574	13	14	τ	τ	X
ejpam-574	13	15	)	)	PUNCT
ejpam-574	13	16	is	be	AUX
ejpam-574	13	17	called	call	VERB
ejpam-574	13	18	a	a	DET
ejpam-574	13	19	topological	topological	ADJ
ejpam-574	13	20	ideal	ideal	NOUN
ejpam-574	13	21	[	[	X
ejpam-574	13	22	6	6	NUM
ejpam-574	13	23	]	]	PUNCT
ejpam-574	13	24	if	if	SCONJ
ejpam-574	13	25	it	it	PRON
ejpam-574	13	26	satisfies	satisfy	VERB
ejpam-574	13	27	the	the	DET
ejpam-574	13	28	following	follow	VERB
ejpam-574	13	29	two	two	NUM
ejpam-574	13	30	conditions	condition	NOUN
ejpam-574	13	31	:	:	PUNCT
ejpam-574	14	1	1	1	X
ejpam-574	14	2	.	.	X
ejpam-574	15	1	if	if	SCONJ
ejpam-574	15	2	a	a	DET
ejpam-574	15	3	∈	∈	X
ejpam-574	15	4	i	i	PRON
ejpam-574	15	5	and	and	CCONJ
ejpam-574	15	6	b	b	PROPN
ejpam-574	15	7	⊂	⊂	PROPN
ejpam-574	15	8	a	a	PRON
ejpam-574	15	9	implies	imply	VERB
ejpam-574	15	10	b	b	X
ejpam-574	15	11	∈	∈	X
ejpam-574	15	12	i	i	PRON
ejpam-574	15	13	(	(	PUNCT
ejpam-574	15	14	heredity	heredity	NOUN
ejpam-574	15	15	)	)	PUNCT
ejpam-574	15	16	2	2	NUM
ejpam-574	15	17	.	.	X
ejpam-574	16	1	if	if	SCONJ
ejpam-574	16	2	a	a	DET
ejpam-574	16	3	∈	∈	X
ejpam-574	16	4	i	i	PRON
ejpam-574	16	5	and	and	CCONJ
ejpam-574	16	6	b	b	X
ejpam-574	16	7	∈	∈	PROPN
ejpam-574	16	8	i	i	PRON
ejpam-574	16	9	,	,	PUNCT
ejpam-574	16	10	then	then	ADV
ejpam-574	16	11	a	a	DET
ejpam-574	16	12	∪	∪	X
ejpam-574	16	13	b	b	X
ejpam-574	16	14	∈	∈	NOUN
ejpam-574	16	15	i	i	PRON
ejpam-574	16	16	(	(	PUNCT
ejpam-574	16	17	finite	finite	PROPN
ejpam-574	16	18	additivity	additivity	NOUN
ejpam-574	16	19	)	)	PUNCT
ejpam-574	16	20	if	if	SCONJ
ejpam-574	16	21	a	a	PRON
ejpam-574	16	22	is	be	AUX
ejpam-574	16	23	a	a	DET
ejpam-574	16	24	subset	subset	NOUN
ejpam-574	16	25	of	of	ADP
ejpam-574	16	26	a	a	DET
ejpam-574	16	27	topological	topological	ADJ
ejpam-574	16	28	space	space	NOUN
ejpam-574	16	29	(	(	PUNCT
ejpam-574	16	30	x	x	X
ejpam-574	16	31	,	,	PUNCT
ejpam-574	16	32	τ	τ	PROPN
ejpam-574	16	33	)	)	PUNCT
ejpam-574	16	34	,	,	PUNCT
ejpam-574	16	35	cl(a	cl(a	NUM
ejpam-574	16	36	)	)	PUNCT
ejpam-574	16	37	and	and	CCONJ
ejpam-574	16	38	int(a	int(a	PROPN
ejpam-574	16	39	)	)	PUNCT
ejpam-574	16	40	denote	denote	VERB
ejpam-574	16	41	the	the	DET
ejpam-574	16	42	closure	closure	NOUN
ejpam-574	16	43	of	of	ADP
ejpam-574	16	44	a	a	PRON
ejpam-574	16	45	and	and	CCONJ
ejpam-574	16	46	the	the	DET
ejpam-574	16	47	interior	interior	NOUN
ejpam-574	16	48	of	of	ADP
ejpam-574	16	49	a	a	PRON
ejpam-574	16	50	,	,	PUNCT
ejpam-574	16	51	respectively	respectively	ADV
ejpam-574	16	52	.	.	PUNCT
ejpam-574	17	1	let	let	VERB
ejpam-574	17	2	a⊂b⊂x	a⊂b⊂x	PROPN
ejpam-574	17	3	.	.	PUNCT
ejpam-574	18	1	then	then	ADV
ejpam-574	18	2	clb(a	clb(a	NUM
ejpam-574	18	3	)	)	PUNCT
ejpam-574	18	4	(	(	PUNCT
ejpam-574	18	5	resp	resp	NOUN
ejpam-574	18	6	.	.	PUNCT
ejpam-574	19	1	intb(a	intb(a	ADJ
ejpam-574	19	2	)	)	PUNCT
ejpam-574	19	3	)	)	PUNCT
ejpam-574	19	4	denotes	denote	VERB
ejpam-574	19	5	closure	closure	NOUN
ejpam-574	19	6	of	of	ADP
ejpam-574	19	7	a	a	DET
ejpam-574	19	8	(	(	PUNCT
ejpam-574	19	9	resp	resp	NOUN
ejpam-574	19	10	.	.	PUNCT
ejpam-574	20	1	interior	interior	NOUN
ejpam-574	20	2	of	of	ADP
ejpam-574	20	3	a	a	PRON
ejpam-574	20	4	)	)	PUNCT
ejpam-574	20	5	with	with	ADP
ejpam-574	20	6	respect	respect	NOUN
ejpam-574	20	7	to	to	ADP
ejpam-574	20	8	b.	b.	PROPN
ejpam-574	20	9	in	in	ADP
ejpam-574	20	10	1963	1963	NUM
ejpam-574	20	11	,	,	PUNCT
ejpam-574	20	12	levine	levine	PROPN
ejpam-574	21	1	[	[	X
ejpam-574	21	2	7	7	NUM
ejpam-574	21	3	]	]	PUNCT
ejpam-574	21	4	introduced	introduce	VERB
ejpam-574	21	5	the	the	DET
ejpam-574	21	6	concept	concept	NOUN
ejpam-574	21	7	of	of	ADP
ejpam-574	21	8	generalized	generalized	ADJ
ejpam-574	21	9	closed	closed	ADJ
ejpam-574	21	10	sets	set	NOUN
ejpam-574	21	11	.	.	PUNCT
ejpam-574	22	1	this	this	DET
ejpam-574	22	2	notion	notion	NOUN
ejpam-574	22	3	has	have	AUX
ejpam-574	22	4	been	be	AUX
ejpam-574	22	5	studied	study	VERB
ejpam-574	22	6	extensively	extensively	ADV
ejpam-574	22	7	in	in	ADP
ejpam-574	22	8	recent	recent	ADJ
ejpam-574	22	9	years	year	NOUN
ejpam-574	22	10	by	by	ADP
ejpam-574	22	11	many	many	ADJ
ejpam-574	22	12	topologists	topologist	NOUN
ejpam-574	22	13	.	.	PUNCT
ejpam-574	23	1	a	a	DET
ejpam-574	23	2	subset	subset	NOUN
ejpam-574	23	3	a	a	PRON
ejpam-574	23	4	of	of	ADP
ejpam-574	23	5	a	a	DET
ejpam-574	23	6	topological	topological	ADJ
ejpam-574	23	7	space	space	NOUN
ejpam-574	23	8	(	(	PUNCT
ejpam-574	23	9	x	x	X
ejpam-574	23	10	,	,	PUNCT
ejpam-574	23	11	τ	τ	X
ejpam-574	23	12	)	)	PUNCT
ejpam-574	23	13	is	be	AUX
ejpam-574	23	14	said	say	VERB
ejpam-574	23	15	to	to	PART
ejpam-574	23	16	be	be	AUX
ejpam-574	23	17	generalized	generalize	VERB
ejpam-574	23	18	closed	close	VERB
ejpam-574	23	19	(	(	PUNCT
ejpam-574	23	20	briefly	briefly	NOUN
ejpam-574	23	21	g	g	NOUN
ejpam-574	23	22	-	-	PUNCT
ejpam-574	23	23	closed	closed	ADJ
ejpam-574	23	24	)	)	PUNCT
ejpam-574	23	25	if	if	SCONJ
ejpam-574	23	26	cl(a	cl(a	NUM
ejpam-574	23	27	)	)	PUNCT
ejpam-574	24	1	⊂	⊂	PROPN
ejpam-574	24	2	u	u	NOUN
ejpam-574	24	3	whenever	whenever	SCONJ
ejpam-574	24	4	a	a	DET
ejpam-574	24	5	⊂	⊂	PROPN
ejpam-574	24	6	u	u	NOUN
ejpam-574	24	7	and	and	CCONJ
ejpam-574	24	8	u	u	NOUN
ejpam-574	24	9	is	be	AUX
ejpam-574	24	10	open	open	ADJ
ejpam-574	24	11	in	in	ADP
ejpam-574	24	12	(	(	PUNCT
ejpam-574	24	13	x	x	INTJ
ejpam-574	24	14	,	,	PUNCT
ejpam-574	24	15	τ	τ	PROPN
ejpam-574	24	16	)	)	PUNCT
ejpam-574	24	17	.	.	PUNCT
ejpam-574	25	1	in	in	ADP
ejpam-574	25	2	this	this	DET
ejpam-574	25	3	paper	paper	NOUN
ejpam-574	25	4	,	,	PUNCT
ejpam-574	25	5	we	we	PRON
ejpam-574	25	6	introduce	introduce	VERB
ejpam-574	25	7	and	and	CCONJ
ejpam-574	25	8	study	study	VERB
ejpam-574	25	9	the	the	DET
ejpam-574	25	10	concept	concept	NOUN
ejpam-574	25	11	of	of	ADP
ejpam-574	25	12	g	g	NOUN
ejpam-574	25	13	-	-	PUNCT
ejpam-574	25	14	closed	close	VERB
ejpam-574	25	15	sets	set	NOUN
ejpam-574	25	16	with	with	ADP
ejpam-574	25	17	respect	respect	NOUN
ejpam-574	25	18	to	to	ADP
ejpam-574	25	19	an	an	DET
ejpam-574	25	20	ideal	ideal	NOUN
ejpam-574	25	21	,	,	PUNCT
ejpam-574	25	22	which	which	PRON
ejpam-574	25	23	is	be	AUX
ejpam-574	25	24	the	the	DET
ejpam-574	25	25	extension	extension	NOUN
ejpam-574	25	26	of	of	ADP
ejpam-574	25	27	the	the	DET
ejpam-574	25	28	concept	concept	NOUN
ejpam-574	25	29	of	of	ADP
ejpam-574	25	30	g	g	NOUN
ejpam-574	25	31	-	-	PUNCT
ejpam-574	25	32	closed	close	VERB
ejpam-574	25	33	sets	set	NOUN
ejpam-574	25	34	.	.	PUNCT
ejpam-574	26	1	∗corresponding	∗corresponde	VERB
ejpam-574	26	2	author	author	NOUN
ejpam-574	26	3	.	.	PUNCT
ejpam-574	27	1	email	email	NOUN
ejpam-574	27	2	addresses	address	NOUN
ejpam-574	27	3	:	:	PUNCT
ejpam-574	27	4	jafaripersia	jafaripersia	PROPN
ejpam-574	27	5	�	�	PROPN
ejpam-574	27	6	gmail	gmail	NOUN
ejpam-574	27	7	.	.	PUNCT
ejpam-574	28	1	om	om	PROPN
ejpam-574	28	2	(	(	PUNCT
ejpam-574	28	3	s.	s.	PROPN
ejpam-574	28	4	jafari	jafari	PROPN
ejpam-574	28	5	)	)	PUNCT
ejpam-574	28	6	,	,	PUNCT
ejpam-574	28	7	nrajesh_topology	nrajesh_topology	NOUN
ejpam-574	28	8	�	�	PROPN
ejpam-574	28	9	yahoo	yahoo	PROPN
ejpam-574	28	10	.	.	PUNCT
ejpam-574	29	1	o.in	o.in	PROPN
ejpam-574	29	2	(	(	PUNCT
ejpam-574	29	3	n.	n.	PROPN
ejpam-574	29	4	rajesh	rajesh	PROPN
ejpam-574	29	5	)	)	PUNCT
ejpam-574	29	6	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-574	30	1	147	147	NUM
ejpam-574	30	2	c	c	X
ejpam-574	30	3	©	©	PROPN
ejpam-574	30	4	2011	2011	NUM
ejpam-574	30	5	ejpam	ejpam	VERB
ejpam-574	30	6	all	all	DET
ejpam-574	30	7	rights	right	NOUN
ejpam-574	30	8	reserved	reserve	VERB
ejpam-574	30	9	.	.	PUNCT
ejpam-574	31	1	s.	s.	PROPN
ejpam-574	31	2	jafari	jafari	PROPN
ejpam-574	31	3	,	,	PUNCT
ejpam-574	31	4	n.	n.	PROPN
ejpam-574	31	5	rajesh	rajesh	PROPN
ejpam-574	31	6	/	/	SYM
ejpam-574	31	7	eur	eur	PROPN
ejpam-574	31	8	.	.	PUNCT
ejpam-574	32	1	j.	j.	PROPN
ejpam-574	32	2	pure	pure	PROPN
ejpam-574	32	3	appl	appl	PROPN
ejpam-574	32	4	.	.	PROPN
ejpam-574	32	5	math	math	PROPN
ejpam-574	32	6	,	,	PUNCT
ejpam-574	32	7	4	4	NUM
ejpam-574	32	8	(	(	PUNCT
ejpam-574	32	9	2011	2011	NUM
ejpam-574	32	10	)	)	PUNCT
ejpam-574	32	11	,	,	PUNCT
ejpam-574	32	12	147	147	NUM
ejpam-574	32	13	-	-	SYM
ejpam-574	32	14	151	151	NUM
ejpam-574	32	15	148	148	NUM
ejpam-574	32	16	2	2	NUM
ejpam-574	32	17	.	.	PUNCT
ejpam-574	32	18	generalized	generalize	VERB
ejpam-574	32	19	closed	close	VERB
ejpam-574	32	20	sets	set	NOUN
ejpam-574	32	21	with	with	ADP
ejpam-574	32	22	respect	respect	NOUN
ejpam-574	32	23	to	to	ADP
ejpam-574	32	24	an	an	DET
ejpam-574	32	25	ideal	ideal	ADJ
ejpam-574	32	26	definition	definition	NOUN
ejpam-574	32	27	1	1	NUM
ejpam-574	32	28	.	.	PUNCT
ejpam-574	33	1	let	let	AUX
ejpam-574	33	2	(	(	PUNCT
ejpam-574	33	3	x	x	X
ejpam-574	33	4	,	,	PUNCT
ejpam-574	33	5	τ	τ	X
ejpam-574	33	6	)	)	PUNCT
ejpam-574	33	7	be	be	VERB
ejpam-574	33	8	a	a	DET
ejpam-574	33	9	topological	topological	ADJ
ejpam-574	33	10	space	space	NOUN
ejpam-574	34	1	and	and	CCONJ
ejpam-574	34	2	i	i	PRON
ejpam-574	34	3	be	be	VERB
ejpam-574	34	4	an	an	DET
ejpam-574	34	5	ideal	ideal	NOUN
ejpam-574	34	6	on	on	ADP
ejpam-574	34	7	x.	x.	PROPN
ejpam-574	34	8	a	a	DET
ejpam-574	34	9	subset	subset	NOUN
ejpam-574	34	10	a	a	PRON
ejpam-574	34	11	of	of	ADP
ejpam-574	34	12	x	x	SYM
ejpam-574	34	13	is	be	AUX
ejpam-574	34	14	said	say	VERB
ejpam-574	34	15	to	to	PART
ejpam-574	34	16	be	be	AUX
ejpam-574	34	17	generalized	generalize	VERB
ejpam-574	34	18	closed	close	VERB
ejpam-574	34	19	with	with	ADP
ejpam-574	34	20	respect	respect	NOUN
ejpam-574	34	21	to	to	ADP
ejpam-574	34	22	an	an	DET
ejpam-574	34	23	ideal	ideal	NOUN
ejpam-574	34	24	(	(	PUNCT
ejpam-574	34	25	briefly	briefly	ADV
ejpam-574	34	26	ig	ig	NOUN
ejpam-574	34	27	-	-	PUNCT
ejpam-574	34	28	closed	closed	ADJ
ejpam-574	34	29	)	)	PUNCT
ejpam-574	35	1	if	if	SCONJ
ejpam-574	35	2	and	and	CCONJ
ejpam-574	35	3	only	only	ADV
ejpam-574	35	4	if	if	SCONJ
ejpam-574	35	5	cl(a	cl(a	NUM
ejpam-574	35	6	)	)	PUNCT
ejpam-574	35	7	–	–	PUNCT
ejpam-574	36	1	b	b	X
ejpam-574	36	2	∈	∈	NOUN
ejpam-574	37	1	i	i	PRON
ejpam-574	37	2	,	,	PUNCT
ejpam-574	37	3	whenever	whenever	SCONJ
ejpam-574	37	4	a	a	DET
ejpam-574	37	5	⊂	⊂	PROPN
ejpam-574	37	6	b	b	PROPN
ejpam-574	37	7	and	and	CCONJ
ejpam-574	37	8	b	b	PROPN
ejpam-574	37	9	is	be	AUX
ejpam-574	37	10	open	open	ADJ
ejpam-574	37	11	.	.	PUNCT
ejpam-574	38	1	remark	remark	NOUN
ejpam-574	38	2	1	1	NUM
ejpam-574	38	3	.	.	PUNCT
ejpam-574	39	1	every	every	DET
ejpam-574	39	2	g	g	NOUN
ejpam-574	39	3	-	-	PUNCT
ejpam-574	39	4	closed	close	VERB
ejpam-574	39	5	set	set	NOUN
ejpam-574	39	6	is	be	AUX
ejpam-574	39	7	ig	ig	NOUN
ejpam-574	39	8	-	-	ADJ
ejpam-574	39	9	closed	closed	ADJ
ejpam-574	39	10	,	,	PUNCT
ejpam-574	39	11	but	but	CCONJ
ejpam-574	39	12	the	the	DET
ejpam-574	39	13	converse	converse	NOUN
ejpam-574	39	14	need	need	AUX
ejpam-574	39	15	not	not	PART
ejpam-574	39	16	be	be	AUX
ejpam-574	39	17	true	true	ADJ
ejpam-574	39	18	,	,	PUNCT
ejpam-574	39	19	as	as	SCONJ
ejpam-574	39	20	this	this	PRON
ejpam-574	39	21	may	may	AUX
ejpam-574	39	22	be	be	AUX
ejpam-574	39	23	seen	see	VERB
ejpam-574	39	24	from	from	ADP
ejpam-574	39	25	the	the	DET
ejpam-574	39	26	following	follow	VERB
ejpam-574	39	27	example	example	NOUN
ejpam-574	39	28	.	.	PUNCT
ejpam-574	40	1	example	example	NOUN
ejpam-574	41	1	1	1	NUM
ejpam-574	41	2	.	.	PUNCT
ejpam-574	41	3	let	let	VERB
ejpam-574	41	4	x={a	x={a	PROPN
ejpam-574	41	5	,	,	PUNCT
ejpam-574	41	6	b	b	PROPN
ejpam-574	41	7	,	,	PUNCT
ejpam-574	41	8	c	c	NOUN
ejpam-574	41	9	}	}	PUNCT
ejpam-574	41	10	with	with	ADP
ejpam-574	41	11	topology	topology	NOUN
ejpam-574	41	12	τ={∅	τ={∅	PROPN
ejpam-574	41	13	,	,	PUNCT
ejpam-574	41	14	{	{	PUNCT
ejpam-574	41	15	a	a	X
ejpam-574	41	16	}	}	PUNCT
ejpam-574	41	17	,	,	PUNCT
ejpam-574	41	18	{	{	PUNCT
ejpam-574	41	19	a	a	X
ejpam-574	41	20	,	,	PUNCT
ejpam-574	41	21	c	c	NOUN
ejpam-574	41	22	}	}	PUNCT
ejpam-574	41	23	,	,	PUNCT
ejpam-574	41	24	x	x	NOUN
ejpam-574	41	25	}	}	PUNCT
ejpam-574	41	26	and	and	CCONJ
ejpam-574	41	27	i=	i=	PROPN
ejpam-574	41	28	{	{	PUNCT
ejpam-574	41	29	;	;	PUNCT
ejpam-574	41	30	,	,	PUNCT
ejpam-574	41	31	{	{	PUNCT
ejpam-574	41	32	b	b	NOUN
ejpam-574	41	33	}	}	PUNCT
ejpam-574	41	34	,	,	PUNCT
ejpam-574	41	35	{	{	PUNCT
ejpam-574	41	36	c	c	X
ejpam-574	41	37	}	}	PUNCT
ejpam-574	41	38	,	,	PUNCT
ejpam-574	41	39	{	{	PUNCT
ejpam-574	41	40	b	b	X
ejpam-574	41	41	,	,	PUNCT
ejpam-574	41	42	c	c	NOUN
ejpam-574	41	43	}	}	PUNCT
ejpam-574	41	44	}	}	PUNCT
ejpam-574	41	45	.	.	PUNCT
ejpam-574	42	1	clearly	clearly	ADV
ejpam-574	42	2	,	,	PUNCT
ejpam-574	42	3	the	the	DET
ejpam-574	42	4	set	set	NOUN
ejpam-574	42	5	{	{	PUNCT
ejpam-574	42	6	c	c	NOUN
ejpam-574	42	7	}	}	PUNCT
ejpam-574	42	8	is	be	AUX
ejpam-574	42	9	ig	ig	PRON
ejpam-574	42	10	-	-	ADJ
ejpam-574	42	11	closed	closed	ADJ
ejpam-574	42	12	but	but	CCONJ
ejpam-574	42	13	not	not	PART
ejpam-574	42	14	g	g	NOUN
ejpam-574	42	15	-	-	PUNCT
ejpam-574	42	16	closed	closed	ADJ
ejpam-574	42	17	in	in	ADP
ejpam-574	42	18	(	(	PUNCT
ejpam-574	42	19	x	x	INTJ
ejpam-574	42	20	,	,	PUNCT
ejpam-574	42	21	τ	τ	PROPN
ejpam-574	42	22	)	)	PUNCT
ejpam-574	42	23	.	.	PUNCT
ejpam-574	43	1	the	the	DET
ejpam-574	43	2	following	follow	VERB
ejpam-574	43	3	theorem	theorem	NOUN
ejpam-574	43	4	gives	give	VERB
ejpam-574	43	5	a	a	DET
ejpam-574	43	6	characterization	characterization	NOUN
ejpam-574	43	7	of	of	ADP
ejpam-574	43	8	ig	ig	NOUN
ejpam-574	43	9	-	-	PUNCT
ejpam-574	43	10	closed	close	VERB
ejpam-574	43	11	sets	set	NOUN
ejpam-574	43	12	.	.	PUNCT
ejpam-574	44	1	theorem	theorem	NOUN
ejpam-574	44	2	1	1	NUM
ejpam-574	44	3	.	.	PUNCT
ejpam-574	45	1	a	a	DET
ejpam-574	45	2	set	set	NOUN
ejpam-574	45	3	a	a	PRON
ejpam-574	45	4	is	be	AUX
ejpam-574	45	5	ig	ig	NOUN
ejpam-574	45	6	-	-	VERB
ejpam-574	45	7	closed	closed	ADJ
ejpam-574	45	8	in	in	ADP
ejpam-574	45	9	(	(	PUNCT
ejpam-574	45	10	x	x	INTJ
ejpam-574	45	11	,	,	PUNCT
ejpam-574	45	12	τ	τ	X
ejpam-574	45	13	)	)	PUNCT
ejpam-574	45	14	if	if	SCONJ
ejpam-574	45	15	and	and	CCONJ
ejpam-574	45	16	only	only	ADV
ejpam-574	45	17	if	if	SCONJ
ejpam-574	45	18	f	f	PROPN
ejpam-574	45	19	⊂	⊂	PROPN
ejpam-574	45	20	cl(a	cl(a	X
ejpam-574	45	21	)	)	PUNCT
ejpam-574	45	22	–	–	PUNCT
ejpam-574	45	23	a	a	PRON
ejpam-574	45	24	and	and	CCONJ
ejpam-574	45	25	f	f	PROPN
ejpam-574	45	26	is	be	AUX
ejpam-574	45	27	closed	close	VERB
ejpam-574	45	28	in	in	ADP
ejpam-574	45	29	x	x	PROPN
ejpam-574	45	30	implies	imply	VERB
ejpam-574	45	31	f	f	PROPN
ejpam-574	45	32	∈	∈	PROPN
ejpam-574	45	33	i.	i.	NOUN
ejpam-574	45	34	proof	proof	PROPN
ejpam-574	45	35	.	.	PUNCT
ejpam-574	46	1	assume	assume	VERB
ejpam-574	46	2	that	that	SCONJ
ejpam-574	46	3	a	a	PRON
ejpam-574	46	4	is	be	AUX
ejpam-574	46	5	ig	ig	PRON
ejpam-574	46	6	-	-	ADJ
ejpam-574	46	7	closed	closed	ADJ
ejpam-574	46	8	.	.	PUNCT
ejpam-574	47	1	let	let	VERB
ejpam-574	47	2	f	f	PROPN
ejpam-574	47	3	⊂	⊂	PROPN
ejpam-574	47	4	cl(a	cl(a	X
ejpam-574	47	5	)	)	PUNCT
ejpam-574	47	6	–	–	PUNCT
ejpam-574	47	7	a.	a.	NOUN
ejpam-574	47	8	suppose	suppose	VERB
ejpam-574	47	9	f	f	PROPN
ejpam-574	47	10	is	be	AUX
ejpam-574	47	11	closed	closed	ADJ
ejpam-574	47	12	.	.	PUNCT
ejpam-574	48	1	then	then	ADV
ejpam-574	48	2	a	a	DET
ejpam-574	48	3	⊂	⊂	PROPN
ejpam-574	48	4	x	x	X
ejpam-574	48	5	–	–	PUNCT
ejpam-574	48	6	f.	f.	PROPN
ejpam-574	48	7	by	by	ADP
ejpam-574	48	8	our	our	PRON
ejpam-574	48	9	assumption	assumption	NOUN
ejpam-574	48	10	,	,	PUNCT
ejpam-574	48	11	cl(a	cl(a	NUM
ejpam-574	48	12	)	)	PUNCT
ejpam-574	48	13	–	–	PUNCT
ejpam-574	48	14	(	(	PUNCT
ejpam-574	48	15	x	x	X
ejpam-574	48	16	–	–	PUNCT
ejpam-574	48	17	f	f	X
ejpam-574	48	18	)	)	PUNCT
ejpam-574	48	19	∈	∈	PROPN
ejpam-574	48	20	i.	i.	NOUN
ejpam-574	48	21	but	but	CCONJ
ejpam-574	48	22	f	f	PROPN
ejpam-574	48	23	⊂	⊂	PROPN
ejpam-574	48	24	cl(a	cl(a	X
ejpam-574	48	25	)	)	PUNCT
ejpam-574	48	26	–	–	PUNCT
ejpam-574	48	27	(	(	PUNCT
ejpam-574	48	28	x	x	X
ejpam-574	48	29	−	−	NOUN
ejpam-574	48	30	f	f	X
ejpam-574	48	31	)	)	PUNCT
ejpam-574	48	32	and	and	CCONJ
ejpam-574	48	33	hence	hence	ADV
ejpam-574	48	34	f	f	PROPN
ejpam-574	48	35	∈	∈	PROPN
ejpam-574	48	36	i.	i.	NOUN
ejpam-574	48	37	conversely	conversely	ADV
ejpam-574	48	38	,	,	PUNCT
ejpam-574	48	39	assume	assume	VERB
ejpam-574	48	40	that	that	SCONJ
ejpam-574	48	41	f	f	PROPN
ejpam-574	48	42	⊂	⊂	PROPN
ejpam-574	48	43	cl(a	cl(a	X
ejpam-574	48	44	)	)	PUNCT
ejpam-574	48	45	–	–	PUNCT
ejpam-574	48	46	a	a	PRON
ejpam-574	48	47	and	and	CCONJ
ejpam-574	48	48	f	f	PROPN
ejpam-574	48	49	is	be	AUX
ejpam-574	48	50	closed	close	VERB
ejpam-574	48	51	in	in	ADP
ejpam-574	48	52	x	x	PROPN
ejpam-574	48	53	implies	imply	VERB
ejpam-574	48	54	that	that	SCONJ
ejpam-574	48	55	f	f	PROPN
ejpam-574	48	56	∈	∈	PROPN
ejpam-574	48	57	i.	i.	NOUN
ejpam-574	48	58	suppose	suppose	VERB
ejpam-574	48	59	a	a	DET
ejpam-574	48	60	⊂	⊂	PROPN
ejpam-574	48	61	u	u	NOUN
ejpam-574	48	62	and	and	CCONJ
ejpam-574	48	63	u	u	NOUN
ejpam-574	48	64	is	be	AUX
ejpam-574	48	65	open	open	ADJ
ejpam-574	48	66	.	.	PUNCT
ejpam-574	49	1	then	then	ADV
ejpam-574	49	2	cl(a	cl(a	PUNCT
ejpam-574	49	3	)	)	PUNCT
ejpam-574	49	4	–	–	PUNCT
ejpam-574	49	5	u	u	NOUN
ejpam-574	49	6	=	=	NOUN
ejpam-574	49	7	cl(a	cl(a	X
ejpam-574	49	8	)	)	PUNCT
ejpam-574	49	9	∩	∩	NOUN
ejpam-574	49	10	(	(	PUNCT
ejpam-574	49	11	x	x	X
ejpam-574	49	12	–	–	PUNCT
ejpam-574	49	13	u	u	NOUN
ejpam-574	49	14	)	)	PUNCT
ejpam-574	49	15	is	be	AUX
ejpam-574	49	16	a	a	DET
ejpam-574	49	17	closed	closed	ADJ
ejpam-574	49	18	set	set	NOUN
ejpam-574	49	19	in	in	ADP
ejpam-574	49	20	x	x	NOUN
ejpam-574	49	21	,	,	PUNCT
ejpam-574	49	22	that	that	PRON
ejpam-574	49	23	is	be	AUX
ejpam-574	49	24	contained	contain	VERB
ejpam-574	49	25	in	in	ADP
ejpam-574	49	26	cl(a	cl(a	NUM
ejpam-574	49	27	)	)	PUNCT
ejpam-574	49	28	–	–	PUNCT
ejpam-574	49	29	a.	a.	NOUN
ejpam-574	49	30	by	by	ADP
ejpam-574	49	31	assumption	assumption	NOUN
ejpam-574	49	32	,	,	PUNCT
ejpam-574	49	33	cl(a	cl(a	NUM
ejpam-574	49	34	)	)	PUNCT
ejpam-574	49	35	–	–	PUNCT
ejpam-574	49	36	u	u	PROPN
ejpam-574	49	37	∈	∈	PROPN
ejpam-574	49	38	i.	i.	NOUN
ejpam-574	49	39	this	this	PRON
ejpam-574	49	40	implies	imply	VERB
ejpam-574	49	41	that	that	SCONJ
ejpam-574	49	42	a	a	PRON
ejpam-574	49	43	is	be	AUX
ejpam-574	49	44	ig	ig	PRON
ejpam-574	49	45	-	-	ADJ
ejpam-574	49	46	closed	closed	ADJ
ejpam-574	49	47	.	.	PUNCT
ejpam-574	50	1	theorem	theorem	NOUN
ejpam-574	50	2	2	2	NUM
ejpam-574	50	3	.	.	PUNCT
ejpam-574	51	1	if	if	SCONJ
ejpam-574	51	2	a	a	PRON
ejpam-574	51	3	and	and	CCONJ
ejpam-574	51	4	b	b	NOUN
ejpam-574	51	5	are	be	AUX
ejpam-574	51	6	ig	ig	PRON
ejpam-574	51	7	-	-	ADJ
ejpam-574	51	8	closed	closed	ADJ
ejpam-574	51	9	sets	set	NOUN
ejpam-574	51	10	of	of	ADP
ejpam-574	51	11	(	(	PUNCT
ejpam-574	51	12	x	x	INTJ
ejpam-574	51	13	,	,	PUNCT
ejpam-574	51	14	τ	τ	PROPN
ejpam-574	51	15	)	)	PUNCT
ejpam-574	51	16	,	,	PUNCT
ejpam-574	51	17	then	then	ADV
ejpam-574	51	18	their	their	PRON
ejpam-574	51	19	union	union	NOUN
ejpam-574	51	20	a	a	DET
ejpam-574	51	21	∪	∪	NOUN
ejpam-574	51	22	b	b	NOUN
ejpam-574	51	23	is	be	AUX
ejpam-574	51	24	also	also	ADV
ejpam-574	51	25	ig	ig	PROPN
ejpam-574	51	26	-	-	ADJ
ejpam-574	51	27	closed	closed	ADJ
ejpam-574	51	28	.	.	PUNCT
ejpam-574	52	1	proof	proof	NOUN
ejpam-574	52	2	.	.	PUNCT
ejpam-574	53	1	suppose	suppose	VERB
ejpam-574	53	2	a	a	PRON
ejpam-574	53	3	and	and	CCONJ
ejpam-574	53	4	b	b	NOUN
ejpam-574	53	5	are	be	AUX
ejpam-574	53	6	ig	ig	PRON
ejpam-574	53	7	-	-	ADJ
ejpam-574	53	8	closed	closed	ADJ
ejpam-574	53	9	sets	set	NOUN
ejpam-574	53	10	in	in	ADP
ejpam-574	53	11	(	(	PUNCT
ejpam-574	53	12	x	x	INTJ
ejpam-574	53	13	,	,	PUNCT
ejpam-574	53	14	τ	τ	PROPN
ejpam-574	53	15	)	)	PUNCT
ejpam-574	53	16	.	.	PUNCT
ejpam-574	54	1	if	if	SCONJ
ejpam-574	54	2	a	a	DET
ejpam-574	54	3	∪	∪	X
ejpam-574	54	4	b	b	NOUN
ejpam-574	54	5	⊂	⊂	PROPN
ejpam-574	54	6	u	u	PROPN
ejpam-574	54	7	and	and	CCONJ
ejpam-574	54	8	u	u	NOUN
ejpam-574	54	9	is	be	AUX
ejpam-574	54	10	open	open	ADJ
ejpam-574	54	11	,	,	PUNCT
ejpam-574	54	12	then	then	ADV
ejpam-574	54	13	a	a	DET
ejpam-574	54	14	⊂	⊂	PROPN
ejpam-574	54	15	u	u	NOUN
ejpam-574	54	16	and	and	CCONJ
ejpam-574	54	17	b	b	PROPN
ejpam-574	54	18	⊂	⊂	PROPN
ejpam-574	54	19	u.	u.	PROPN
ejpam-574	54	20	by	by	ADP
ejpam-574	54	21	assumption	assumption	NOUN
ejpam-574	54	22	,	,	PUNCT
ejpam-574	54	23	cl(a	cl(a	NUM
ejpam-574	54	24	)	)	PUNCT
ejpam-574	54	25	–	–	PUNCT
ejpam-574	54	26	u	u	NOUN
ejpam-574	54	27	∈	∈	PROPN
ejpam-574	54	28	i	i	PRON
ejpam-574	54	29	and	and	CCONJ
ejpam-574	54	30	cl(b	cl(b	NOUN
ejpam-574	54	31	)	)	PUNCT
ejpam-574	54	32	–	–	PUNCT
ejpam-574	54	33	u	u	NOUN
ejpam-574	54	34	∈i	∈i	NOUN
ejpam-574	54	35	and	and	CCONJ
ejpam-574	54	36	hence	hence	ADV
ejpam-574	54	37	cl(a	cl(a	X
ejpam-574	54	38	∪	∪	X
ejpam-574	54	39	b	b	NOUN
ejpam-574	54	40	)	)	PUNCT
ejpam-574	54	41	–	–	PUNCT
ejpam-574	54	42	u	u	NOUN
ejpam-574	54	43	=	=	PUNCT
ejpam-574	54	44	(	(	PUNCT
ejpam-574	54	45	cl(a)–u	cl(a)–u	NOUN
ejpam-574	54	46	)	)	PUNCT
ejpam-574	54	47	∪	∪	NOUN
ejpam-574	54	48	(	(	PUNCT
ejpam-574	54	49	cl(b)–u	cl(b)–u	NOUN
ejpam-574	54	50	)	)	PUNCT
ejpam-574	54	51	∈	∈	PROPN
ejpam-574	54	52	i.	i.	NOUN
ejpam-574	54	53	that	that	PRON
ejpam-574	54	54	is	be	AUX
ejpam-574	54	55	a	a	DET
ejpam-574	54	56	∪	∪	ADJ
ejpam-574	54	57	b	b	NOUN
ejpam-574	54	58	is	be	AUX
ejpam-574	54	59	ig	ig	PRON
ejpam-574	54	60	-	-	ADJ
ejpam-574	54	61	closed	closed	ADJ
ejpam-574	54	62	.	.	PUNCT
ejpam-574	55	1	remark	remark	NOUN
ejpam-574	55	2	2	2	NUM
ejpam-574	55	3	.	.	PUNCT
ejpam-574	56	1	the	the	DET
ejpam-574	56	2	intersection	intersection	NOUN
ejpam-574	56	3	of	of	ADP
ejpam-574	56	4	two	two	NUM
ejpam-574	56	5	ig	ig	PROPN
ejpam-574	56	6	-	-	PUNCT
ejpam-574	56	7	closed	close	VERB
ejpam-574	56	8	sets	set	NOUN
ejpam-574	56	9	need	need	AUX
ejpam-574	56	10	not	not	PART
ejpam-574	56	11	be	be	AUX
ejpam-574	56	12	an	an	DET
ejpam-574	56	13	ig	ig	NOUN
ejpam-574	56	14	-	-	PUNCT
ejpam-574	56	15	closed	closed	ADJ
ejpam-574	56	16	as	as	SCONJ
ejpam-574	56	17	shown	show	VERB
ejpam-574	56	18	by	by	ADP
ejpam-574	56	19	the	the	DET
ejpam-574	56	20	following	follow	VERB
ejpam-574	56	21	example	example	NOUN
ejpam-574	56	22	.	.	PUNCT
ejpam-574	57	1	example	example	NOUN
ejpam-574	58	1	2	2	NUM
ejpam-574	58	2	.	.	PUNCT
ejpam-574	58	3	let	let	VERB
ejpam-574	58	4	x	x	PUNCT
ejpam-574	58	5	=	=	PRON
ejpam-574	58	6	{	{	PUNCT
ejpam-574	58	7	a	a	PRON
ejpam-574	58	8	,	,	PUNCT
ejpam-574	58	9	b	b	NOUN
ejpam-574	58	10	,	,	PUNCT
ejpam-574	58	11	c	c	NOUN
ejpam-574	58	12	}	}	PUNCT
ejpam-574	58	13	with	with	ADP
ejpam-574	58	14	topology	topology	NOUN
ejpam-574	58	15	τ	τ	X
ejpam-574	58	16	=	=	SYM
ejpam-574	58	17	{	{	PUNCT
ejpam-574	58	18	∅	∅	NOUN
ejpam-574	58	19	,	,	PUNCT
ejpam-574	58	20	{	{	PUNCT
ejpam-574	58	21	b	b	NOUN
ejpam-574	58	22	}	}	PUNCT
ejpam-574	58	23	,	,	PUNCT
ejpam-574	58	24	x	x	NOUN
ejpam-574	58	25	}	}	PUNCT
ejpam-574	58	26	.	.	PUNCT
ejpam-574	59	1	if	if	SCONJ
ejpam-574	59	2	a	a	PRON
ejpam-574	59	3	=	=	X
ejpam-574	59	4	{	{	PUNCT
ejpam-574	59	5	a	a	PROPN
ejpam-574	59	6	,	,	PUNCT
ejpam-574	59	7	b	b	NOUN
ejpam-574	59	8	}	}	PUNCT
ejpam-574	59	9	,	,	PUNCT
ejpam-574	59	10	b	b	X
ejpam-574	59	11	=	=	PRON
ejpam-574	59	12	{	{	PUNCT
ejpam-574	59	13	b	b	PROPN
ejpam-574	59	14	,	,	PUNCT
ejpam-574	59	15	c	c	NOUN
ejpam-574	59	16	}	}	PUNCT
ejpam-574	59	17	and	and	CCONJ
ejpam-574	59	18	i	i	PRON
ejpam-574	59	19	=	=	PUNCT
ejpam-574	59	20	{	{	PUNCT
ejpam-574	59	21	∅	∅	NOUN
ejpam-574	59	22	}	}	PUNCT
ejpam-574	59	23	,	,	PUNCT
ejpam-574	59	24	then	then	ADV
ejpam-574	59	25	a	a	PRON
ejpam-574	59	26	and	and	CCONJ
ejpam-574	59	27	b	b	NOUN
ejpam-574	59	28	are	be	AUX
ejpam-574	59	29	ig	ig	PRON
ejpam-574	59	30	-	-	VERB
ejpam-574	59	31	closed	closed	ADJ
ejpam-574	59	32	but	but	CCONJ
ejpam-574	59	33	their	their	PRON
ejpam-574	59	34	intersection	intersection	NOUN
ejpam-574	59	35	a	a	DET
ejpam-574	59	36	∩	∩	ADJ
ejpam-574	59	37	b	b	NOUN
ejpam-574	59	38	=	=	SYM
ejpam-574	59	39	{	{	PUNCT
ejpam-574	59	40	b	b	NOUN
ejpam-574	59	41	}	}	PUNCT
ejpam-574	59	42	is	be	AUX
ejpam-574	59	43	not	not	PART
ejpam-574	59	44	ig	ig	NOUN
ejpam-574	59	45	-	-	PUNCT
ejpam-574	59	46	closed	closed	ADJ
ejpam-574	59	47	.	.	PUNCT
ejpam-574	60	1	theorem	theorem	NOUN
ejpam-574	60	2	3	3	NUM
ejpam-574	60	3	.	.	PUNCT
ejpam-574	61	1	if	if	SCONJ
ejpam-574	61	2	a	a	PRON
ejpam-574	61	3	is	be	AUX
ejpam-574	61	4	ig	ig	PRON
ejpam-574	61	5	-	-	PUNCT
ejpam-574	61	6	closed	closed	ADJ
ejpam-574	61	7	and	and	CCONJ
ejpam-574	61	8	a	a	DET
ejpam-574	61	9	⊂	⊂	PROPN
ejpam-574	61	10	b	b	X
ejpam-574	61	11	⊂	⊂	PROPN
ejpam-574	61	12	cl(a	cl(a	X
ejpam-574	61	13	)	)	PUNCT
ejpam-574	61	14	in	in	ADP
ejpam-574	61	15	(	(	PUNCT
ejpam-574	61	16	x	x	INTJ
ejpam-574	61	17	,	,	PUNCT
ejpam-574	61	18	τ	τ	PROPN
ejpam-574	61	19	)	)	PUNCT
ejpam-574	61	20	,	,	PUNCT
ejpam-574	61	21	then	then	ADV
ejpam-574	61	22	b	b	PROPN
ejpam-574	61	23	is	be	AUX
ejpam-574	61	24	ig	ig	PRON
ejpam-574	61	25	-	-	VERB
ejpam-574	61	26	closed	closed	ADJ
ejpam-574	61	27	in	in	ADP
ejpam-574	61	28	(	(	PUNCT
ejpam-574	61	29	x	x	INTJ
ejpam-574	61	30	,	,	PUNCT
ejpam-574	61	31	τ	τ	PROPN
ejpam-574	61	32	)	)	PUNCT
ejpam-574	61	33	.	.	PUNCT
ejpam-574	62	1	proof	proof	NOUN
ejpam-574	62	2	.	.	PUNCT
ejpam-574	63	1	suppose	suppose	VERB
ejpam-574	63	2	a	a	PRON
ejpam-574	63	3	is	be	AUX
ejpam-574	63	4	ig	ig	NOUN
ejpam-574	63	5	-	-	PUNCT
ejpam-574	63	6	closed	closed	ADJ
ejpam-574	63	7	and	and	CCONJ
ejpam-574	63	8	a	a	DET
ejpam-574	63	9	⊂	⊂	PROPN
ejpam-574	63	10	b	b	X
ejpam-574	63	11	⊂	⊂	PROPN
ejpam-574	63	12	cl(a	cl(a	X
ejpam-574	63	13	)	)	PUNCT
ejpam-574	63	14	in	in	ADP
ejpam-574	63	15	(	(	PUNCT
ejpam-574	63	16	x	x	INTJ
ejpam-574	63	17	,	,	PUNCT
ejpam-574	63	18	τ	τ	PROPN
ejpam-574	63	19	)	)	PUNCT
ejpam-574	63	20	.	.	PUNCT
ejpam-574	64	1	suppose	suppose	VERB
ejpam-574	64	2	b⊂	b⊂	PROPN
ejpam-574	64	3	u	u	PROPN
ejpam-574	64	4	and	and	CCONJ
ejpam-574	64	5	u	u	NOUN
ejpam-574	64	6	is	be	AUX
ejpam-574	64	7	open	open	ADJ
ejpam-574	64	8	.	.	PUNCT
ejpam-574	65	1	then	then	ADV
ejpam-574	65	2	a	a	DET
ejpam-574	65	3	⊂	⊂	X
ejpam-574	65	4	u.	u.	PROPN
ejpam-574	65	5	since	since	SCONJ
ejpam-574	65	6	a	a	PRON
ejpam-574	65	7	is	be	AUX
ejpam-574	65	8	ig	ig	PRON
ejpam-574	65	9	-	-	ADJ
ejpam-574	65	10	closed	closed	ADJ
ejpam-574	65	11	,	,	PUNCT
ejpam-574	65	12	we	we	PRON
ejpam-574	65	13	have	have	VERB
ejpam-574	65	14	cl(a)–u	cl(a)–u	PROPN
ejpam-574	65	15	∈	∈	PROPN
ejpam-574	65	16	i.	i.	NOUN
ejpam-574	65	17	now	now	ADV
ejpam-574	65	18	b	b	PROPN
ejpam-574	65	19	⊂	⊂	PROPN
ejpam-574	65	20	cl(a	cl(a	X
ejpam-574	65	21	)	)	PUNCT
ejpam-574	65	22	.	.	PUNCT
ejpam-574	66	1	this	this	PRON
ejpam-574	66	2	implies	imply	VERB
ejpam-574	66	3	that	that	SCONJ
ejpam-574	66	4	cl(b)–u	cl(b)–u	NOUN
ejpam-574	66	5	⊂	⊂	PUNCT
ejpam-574	66	6	cl(a)–u	cl(a)–u	PROPN
ejpam-574	66	7	∈	∈	PROPN
ejpam-574	66	8	i.	i.	NOUN
ejpam-574	66	9	hence	hence	ADV
ejpam-574	66	10	b	b	PROPN
ejpam-574	66	11	is	be	AUX
ejpam-574	66	12	ig	ig	PRON
ejpam-574	66	13	-	-	VERB
ejpam-574	66	14	closed	closed	ADJ
ejpam-574	66	15	in	in	ADP
ejpam-574	66	16	(	(	PUNCT
ejpam-574	66	17	x	x	INTJ
ejpam-574	66	18	,	,	PUNCT
ejpam-574	66	19	τ	τ	PROPN
ejpam-574	66	20	)	)	PUNCT
ejpam-574	66	21	.	.	PUNCT
ejpam-574	67	1	theorem	theorem	ADJ
ejpam-574	67	2	4	4	NUM
ejpam-574	67	3	.	.	PUNCT
ejpam-574	68	1	let	let	VERB
ejpam-574	68	2	a	a	DET
ejpam-574	68	3	⊂	⊂	PROPN
ejpam-574	68	4	y	y	PROPN
ejpam-574	68	5	⊂	⊂	PROPN
ejpam-574	68	6	x	x	X
ejpam-574	68	7	and	and	CCONJ
ejpam-574	68	8	suppose	suppose	VERB
ejpam-574	68	9	that	that	SCONJ
ejpam-574	68	10	a	a	PRON
ejpam-574	68	11	is	be	AUX
ejpam-574	68	12	ig	ig	NOUN
ejpam-574	68	13	-	-	VERB
ejpam-574	68	14	closed	closed	ADJ
ejpam-574	68	15	in	in	ADP
ejpam-574	68	16	(	(	PUNCT
ejpam-574	68	17	x	x	INTJ
ejpam-574	68	18	,	,	PUNCT
ejpam-574	68	19	τ	τ	PROPN
ejpam-574	68	20	)	)	PUNCT
ejpam-574	68	21	.	.	PUNCT
ejpam-574	69	1	then	then	ADV
ejpam-574	69	2	a	a	PRON
ejpam-574	69	3	is	be	AUX
ejpam-574	69	4	ig	ig	PRON
ejpam-574	69	5	-	-	ADJ
ejpam-574	69	6	closed	closed	ADJ
ejpam-574	69	7	relative	relative	ADJ
ejpam-574	69	8	to	to	ADP
ejpam-574	69	9	the	the	DET
ejpam-574	69	10	subspace	subspace	NOUN
ejpam-574	69	11	y	y	PROPN
ejpam-574	69	12	of	of	ADP
ejpam-574	69	13	x	x	PROPN
ejpam-574	69	14	,	,	PUNCT
ejpam-574	69	15	with	with	ADP
ejpam-574	69	16	respect	respect	NOUN
ejpam-574	69	17	to	to	ADP
ejpam-574	69	18	the	the	DET
ejpam-574	69	19	ideal	ideal	NOUN
ejpam-574	69	20	iy	iy	X
ejpam-574	70	1	=	=	PUNCT
ejpam-574	70	2	{	{	PUNCT
ejpam-574	70	3	f	f	PROPN
ejpam-574	70	4	⊂	⊂	PROPN
ejpam-574	70	5	y	y	PROPN
ejpam-574	70	6	:	:	PUNCT
ejpam-574	70	7	f	f	PROPN
ejpam-574	70	8	∈	∈	PROPN
ejpam-574	71	1	i	i	X
ejpam-574	71	2	}	}	PUNCT
ejpam-574	71	3	.	.	PUNCT
ejpam-574	72	1	proof	proof	NOUN
ejpam-574	72	2	.	.	PUNCT
ejpam-574	73	1	suppose	suppose	VERB
ejpam-574	73	2	a	a	DET
ejpam-574	73	3	⊂	⊂	PROPN
ejpam-574	73	4	u	u	PROPN
ejpam-574	73	5	∩	∩	NOUN
ejpam-574	73	6	y	y	PROPN
ejpam-574	73	7	and	and	CCONJ
ejpam-574	73	8	u	u	NOUN
ejpam-574	73	9	is	be	AUX
ejpam-574	73	10	open	open	ADJ
ejpam-574	73	11	in	in	ADP
ejpam-574	73	12	(	(	PUNCT
ejpam-574	73	13	x	x	INTJ
ejpam-574	73	14	,	,	PUNCT
ejpam-574	73	15	τ	τ	PROPN
ejpam-574	73	16	)	)	PUNCT
ejpam-574	73	17	,	,	PUNCT
ejpam-574	73	18	then	then	ADV
ejpam-574	73	19	a	a	DET
ejpam-574	73	20	⊂u	⊂u	NOUN
ejpam-574	73	21	.	.	PUNCT
ejpam-574	74	1	since	since	SCONJ
ejpam-574	74	2	a	a	PRON
ejpam-574	74	3	is	be	AUX
ejpam-574	74	4	ig	ig	NOUN
ejpam-574	74	5	-	-	VERB
ejpam-574	74	6	closed	closed	ADJ
ejpam-574	74	7	in	in	ADP
ejpam-574	74	8	(	(	PUNCT
ejpam-574	74	9	x	x	INTJ
ejpam-574	74	10	,	,	PUNCT
ejpam-574	74	11	τ	τ	PROPN
ejpam-574	74	12	)	)	PUNCT
ejpam-574	74	13	,	,	PUNCT
ejpam-574	74	14	we	we	PRON
ejpam-574	74	15	have	have	VERB
ejpam-574	74	16	cl(a)–u	cl(a)–u	PROPN
ejpam-574	74	17	∈	∈	PROPN
ejpam-574	74	18	i.	i.	NOUN
ejpam-574	74	19	now	now	ADV
ejpam-574	74	20	(	(	PUNCT
ejpam-574	74	21	cl(a)∩y	cl(a)∩y	PROPN
ejpam-574	74	22	)	)	PUNCT
ejpam-574	74	23	–	–	PUNCT
ejpam-574	74	24	(	(	PUNCT
ejpam-574	74	25	u∩y	u∩y	PROPN
ejpam-574	74	26	)	)	PUNCT
ejpam-574	74	27	=	=	PUNCT
ejpam-574	75	1	(	(	PUNCT
ejpam-574	75	2	cl(a)–u)∩y	cl(a)–u)∩y	NOUN
ejpam-574	75	3	∈	∈	PROPN
ejpam-574	76	1	i	i	PRON
ejpam-574	76	2	,	,	PUNCT
ejpam-574	76	3	whenever	whenever	SCONJ
ejpam-574	76	4	a	a	DET
ejpam-574	76	5	⊂	⊂	PROPN
ejpam-574	76	6	u	u	PROPN
ejpam-574	76	7	∩	∩	NOUN
ejpam-574	76	8	y	y	PROPN
ejpam-574	76	9	and	and	CCONJ
ejpam-574	76	10	u	u	NOUN
ejpam-574	76	11	is	be	AUX
ejpam-574	76	12	open	open	ADJ
ejpam-574	76	13	.	.	PUNCT
ejpam-574	77	1	hence	hence	ADV
ejpam-574	77	2	a	a	PRON
ejpam-574	77	3	is	be	AUX
ejpam-574	77	4	ig	ig	PRON
ejpam-574	77	5	-	-	ADJ
ejpam-574	77	6	closed	closed	ADJ
ejpam-574	77	7	relative	relative	ADJ
ejpam-574	77	8	to	to	ADP
ejpam-574	77	9	the	the	DET
ejpam-574	77	10	subspace	subspace	PROPN
ejpam-574	77	11	y.	y.	PROPN
ejpam-574	77	12	s.	s.	PROPN
ejpam-574	77	13	jafari	jafari	PROPN
ejpam-574	77	14	,	,	PUNCT
ejpam-574	77	15	n.	n.	PROPN
ejpam-574	77	16	rajesh	rajesh	PROPN
ejpam-574	77	17	/	/	SYM
ejpam-574	77	18	eur	eur	PROPN
ejpam-574	77	19	.	.	PUNCT
ejpam-574	78	1	j.	j.	PROPN
ejpam-574	78	2	pure	pure	PROPN
ejpam-574	78	3	appl	appl	PROPN
ejpam-574	78	4	.	.	PROPN
ejpam-574	78	5	math	math	PROPN
ejpam-574	78	6	,	,	PUNCT
ejpam-574	78	7	4	4	NUM
ejpam-574	78	8	(	(	PUNCT
ejpam-574	78	9	2011	2011	NUM
ejpam-574	78	10	)	)	PUNCT
ejpam-574	78	11	,	,	PUNCT
ejpam-574	78	12	147	147	NUM
ejpam-574	78	13	-	-	SYM
ejpam-574	78	14	151	151	NUM
ejpam-574	78	15	149	149	NUM
ejpam-574	78	16	theorem	theorem	NOUN
ejpam-574	78	17	5	5	NUM
ejpam-574	78	18	.	.	PUNCT
ejpam-574	78	19	let	let	VERB
ejpam-574	78	20	a	a	DET
ejpam-574	78	21	be	be	AUX
ejpam-574	78	22	an	an	DET
ejpam-574	78	23	ig	ig	NOUN
ejpam-574	78	24	-	-	PUNCT
ejpam-574	78	25	closed	close	VERB
ejpam-574	78	26	set	set	NOUN
ejpam-574	78	27	and	and	CCONJ
ejpam-574	78	28	f	f	PROPN
ejpam-574	78	29	be	be	AUX
ejpam-574	78	30	a	a	DET
ejpam-574	78	31	closed	closed	ADJ
ejpam-574	78	32	set	set	NOUN
ejpam-574	78	33	in	in	ADP
ejpam-574	78	34	(	(	PUNCT
ejpam-574	78	35	x	x	INTJ
ejpam-574	78	36	,	,	PUNCT
ejpam-574	78	37	τ	τ	PROPN
ejpam-574	78	38	)	)	PUNCT
ejpam-574	78	39	,	,	PUNCT
ejpam-574	78	40	then	then	ADV
ejpam-574	78	41	a	a	DET
ejpam-574	78	42	∩	∩	ADJ
ejpam-574	78	43	f	f	X
ejpam-574	78	44	is	be	AUX
ejpam-574	78	45	an	an	DET
ejpam-574	78	46	ig	ig	PROPN
ejpam-574	78	47	-	-	PUNCT
ejpam-574	78	48	closed	close	VERB
ejpam-574	78	49	set	set	NOUN
ejpam-574	78	50	in	in	ADP
ejpam-574	78	51	(	(	PUNCT
ejpam-574	78	52	x	x	INTJ
ejpam-574	78	53	,	,	PUNCT
ejpam-574	78	54	τ	τ	NOUN
ejpam-574	78	55	)	)	PUNCT
ejpam-574	78	56	proof	proof	NOUN
ejpam-574	78	57	.	.	PUNCT
ejpam-574	79	1	let	let	VERB
ejpam-574	79	2	a	a	DET
ejpam-574	79	3	∩	∩	ADJ
ejpam-574	79	4	f	f	X
ejpam-574	79	5	⊂	⊂	PROPN
ejpam-574	79	6	u	u	PROPN
ejpam-574	79	7	and	and	CCONJ
ejpam-574	79	8	u	u	NOUN
ejpam-574	79	9	is	be	AUX
ejpam-574	79	10	open	open	ADJ
ejpam-574	79	11	.	.	PUNCT
ejpam-574	80	1	then	then	ADV
ejpam-574	80	2	a	a	DET
ejpam-574	80	3	⊂	⊂	PROPN
ejpam-574	80	4	u	u	NOUN
ejpam-574	80	5	∪	∪	X
ejpam-574	80	6	(	(	PUNCT
ejpam-574	80	7	x	x	NOUN
ejpam-574	80	8	–	–	PUNCT
ejpam-574	80	9	f	f	NOUN
ejpam-574	80	10	)	)	PUNCT
ejpam-574	80	11	.	.	PUNCT
ejpam-574	81	1	since	since	SCONJ
ejpam-574	81	2	a	a	PRON
ejpam-574	81	3	is	be	AUX
ejpam-574	81	4	ig	ig	PRON
ejpam-574	81	5	-	-	ADJ
ejpam-574	81	6	closed	closed	ADJ
ejpam-574	81	7	,	,	PUNCT
ejpam-574	81	8	we	we	PRON
ejpam-574	81	9	have	have	VERB
ejpam-574	81	10	cl(a)–(u	cl(a)–(u	NOUN
ejpam-574	81	11	∪	∪	NOUN
ejpam-574	81	12	(	(	PUNCT
ejpam-574	81	13	x	x	NOUN
ejpam-574	81	14	–	–	PUNCT
ejpam-574	81	15	f	f	NOUN
ejpam-574	81	16	)	)	PUNCT
ejpam-574	81	17	)	)	PUNCT
ejpam-574	82	1	∈	∈	PROPN
ejpam-574	82	2	i.	i.	NOUN
ejpam-574	82	3	now	now	ADV
ejpam-574	82	4	,	,	PUNCT
ejpam-574	82	5	cl(a	cl(a	X
ejpam-574	82	6	∩	∩	X
ejpam-574	82	7	f	f	X
ejpam-574	82	8	)	)	PUNCT
ejpam-574	82	9	⊂	⊂	PROPN
ejpam-574	82	10	cl(a	cl(a	NUM
ejpam-574	82	11	)	)	PUNCT
ejpam-574	82	12	∩	∩	NOUN
ejpam-574	82	13	f	f	X
ejpam-574	82	14	=	=	SYM
ejpam-574	82	15	(	(	PUNCT
ejpam-574	82	16	cl(a)∩f	cl(a)∩f	PROPN
ejpam-574	82	17	)	)	PUNCT
ejpam-574	82	18	–	–	PUNCT
ejpam-574	82	19	(	(	PUNCT
ejpam-574	82	20	x	x	X
ejpam-574	82	21	–	–	PUNCT
ejpam-574	82	22	f	f	NOUN
ejpam-574	82	23	)	)	PUNCT
ejpam-574	82	24	.	.	PUNCT
ejpam-574	83	1	therefore	therefore	ADV
ejpam-574	83	2	,	,	PUNCT
ejpam-574	83	3	cl(a∩	cl(a∩	PROPN
ejpam-574	83	4	f)−	f)−	PROPN
ejpam-574	83	5	u	u	PROPN
ejpam-574	83	6	⊂	⊂	X
ejpam-574	83	7	(	(	PUNCT
ejpam-574	83	8	cl(a)∩	cl(a)∩	X
ejpam-574	83	9	f)−	f)−	PROPN
ejpam-574	83	10	(	(	PUNCT
ejpam-574	83	11	u	u	NOUN
ejpam-574	83	12	∩	∩	NOUN
ejpam-574	83	13	(	(	PUNCT
ejpam-574	83	14	x	x	SYM
ejpam-574	83	15	−	−	PROPN
ejpam-574	83	16	f	f	X
ejpam-574	83	17	)	)	PUNCT
ejpam-574	83	18	)	)	PUNCT
ejpam-574	84	1	⊂	⊂	PRON
ejpam-574	84	2	cl(a)−	cl(a)−	ADV
ejpam-574	84	3	(	(	PUNCT
ejpam-574	84	4	u	u	NOUN
ejpam-574	84	5	∪	∪	VERB
ejpam-574	84	6	(	(	PUNCT
ejpam-574	84	7	x	x	SYM
ejpam-574	84	8	−	−	PROPN
ejpam-574	84	9	f	f	X
ejpam-574	84	10	)	)	PUNCT
ejpam-574	84	11	)	)	PUNCT
ejpam-574	85	1	∈	∈	PROPN
ejpam-574	86	1	i	i	PRON
ejpam-574	86	2	hence	hence	ADV
ejpam-574	86	3	a	a	DET
ejpam-574	86	4	∩	∩	ADJ
ejpam-574	86	5	f	f	X
ejpam-574	86	6	is	be	AUX
ejpam-574	86	7	ig	ig	PROPN
ejpam-574	86	8	-	-	ADJ
ejpam-574	86	9	closed	closed	ADJ
ejpam-574	86	10	in	in	ADP
ejpam-574	86	11	(	(	PUNCT
ejpam-574	86	12	x	x	INTJ
ejpam-574	86	13	,	,	PUNCT
ejpam-574	86	14	τ	τ	PROPN
ejpam-574	86	15	)	)	PUNCT
ejpam-574	86	16	.	.	PUNCT
ejpam-574	87	1	definition	definition	NOUN
ejpam-574	87	2	2	2	NUM
ejpam-574	87	3	.	.	PUNCT
ejpam-574	88	1	let	let	AUX
ejpam-574	88	2	(	(	PUNCT
ejpam-574	88	3	x	x	X
ejpam-574	88	4	,	,	PUNCT
ejpam-574	88	5	τ	τ	X
ejpam-574	88	6	)	)	PUNCT
ejpam-574	88	7	be	be	VERB
ejpam-574	88	8	a	a	DET
ejpam-574	88	9	topological	topological	ADJ
ejpam-574	88	10	space	space	NOUN
ejpam-574	88	11	and	and	CCONJ
ejpam-574	88	12	a	a	DET
ejpam-574	88	13	be	be	AUX
ejpam-574	88	14	an	an	DET
ejpam-574	88	15	ideal	ideal	NOUN
ejpam-574	88	16	on	on	ADP
ejpam-574	88	17	x.	x.	PROPN
ejpam-574	88	18	a	a	DET
ejpam-574	88	19	subset	subset	NOUN
ejpam-574	88	20	a	a	DET
ejpam-574	88	21	⊂	⊂	PROPN
ejpam-574	88	22	x	x	X
ejpam-574	88	23	is	be	AUX
ejpam-574	88	24	said	say	VERB
ejpam-574	88	25	to	to	PART
ejpam-574	88	26	be	be	AUX
ejpam-574	88	27	generalized	generalize	VERB
ejpam-574	88	28	open	open	ADJ
ejpam-574	88	29	with	with	ADP
ejpam-574	88	30	respect	respect	NOUN
ejpam-574	88	31	to	to	ADP
ejpam-574	88	32	an	an	DET
ejpam-574	88	33	ideal	ideal	NOUN
ejpam-574	88	34	(	(	PUNCT
ejpam-574	88	35	briefly	briefly	ADV
ejpam-574	88	36	ig	ig	NOUN
ejpam-574	88	37	-	-	ADJ
ejpam-574	88	38	open	open	ADJ
ejpam-574	88	39	)	)	PUNCT
ejpam-574	88	40	if	if	SCONJ
ejpam-574	88	41	and	and	CCONJ
ejpam-574	88	42	only	only	ADV
ejpam-574	88	43	if	if	SCONJ
ejpam-574	88	44	x	x	NOUN
ejpam-574	88	45	–	–	PUNCT
ejpam-574	88	46	a	a	PRON
ejpam-574	88	47	is	be	AUX
ejpam-574	88	48	ig	ig	PRON
ejpam-574	88	49	-	-	ADJ
ejpam-574	88	50	closed	closed	ADJ
ejpam-574	88	51	.	.	PUNCT
ejpam-574	89	1	theorem	theorem	VERB
ejpam-574	89	2	6	6	NUM
ejpam-574	89	3	.	.	PUNCT
ejpam-574	90	1	a	a	DET
ejpam-574	90	2	set	set	NOUN
ejpam-574	90	3	a	a	PRON
ejpam-574	90	4	is	be	AUX
ejpam-574	90	5	ig	ig	NOUN
ejpam-574	90	6	-	-	ADJ
ejpam-574	90	7	open	open	ADJ
ejpam-574	90	8	in	in	ADP
ejpam-574	90	9	(	(	PUNCT
ejpam-574	90	10	x	x	INTJ
ejpam-574	90	11	,	,	PUNCT
ejpam-574	90	12	τ	τ	X
ejpam-574	90	13	)	)	PUNCT
ejpam-574	90	14	if	if	SCONJ
ejpam-574	90	15	and	and	CCONJ
ejpam-574	90	16	only	only	ADV
ejpam-574	90	17	if	if	SCONJ
ejpam-574	90	18	f	f	X
ejpam-574	90	19	–	–	PUNCT
ejpam-574	90	20	u	u	X
ejpam-574	90	21	⊂	⊂	PROPN
ejpam-574	90	22	int(a	int(a	PROPN
ejpam-574	90	23	)	)	PUNCT
ejpam-574	90	24	,	,	PUNCT
ejpam-574	90	25	for	for	ADP
ejpam-574	90	26	some	some	DET
ejpam-574	90	27	u	u	NOUN
ejpam-574	90	28	∈	∈	PROPN
ejpam-574	90	29	i	i	PRON
ejpam-574	90	30	,	,	PUNCT
ejpam-574	91	1	whenever	whenever	SCONJ
ejpam-574	91	2	f	f	PROPN
ejpam-574	91	3	⊂	⊂	PROPN
ejpam-574	91	4	a	a	PROPN
ejpam-574	91	5	and	and	CCONJ
ejpam-574	91	6	f	f	PROPN
ejpam-574	91	7	is	be	AUX
ejpam-574	91	8	closed	closed	ADJ
ejpam-574	91	9	.	.	PUNCT
ejpam-574	92	1	proof	proof	NOUN
ejpam-574	92	2	.	.	PUNCT
ejpam-574	93	1	suppose	suppose	VERB
ejpam-574	93	2	a	a	PRON
ejpam-574	93	3	is	be	AUX
ejpam-574	93	4	ig	ig	NOUN
ejpam-574	93	5	-	-	ADJ
ejpam-574	93	6	open	open	ADJ
ejpam-574	93	7	.	.	PUNCT
ejpam-574	94	1	suppose	suppose	VERB
ejpam-574	94	2	f	f	PROPN
ejpam-574	94	3	⊂	⊂	PROPN
ejpam-574	94	4	a	a	PROPN
ejpam-574	94	5	and	and	CCONJ
ejpam-574	94	6	f	f	PROPN
ejpam-574	94	7	is	be	AUX
ejpam-574	94	8	closed	closed	ADJ
ejpam-574	94	9	.	.	PUNCT
ejpam-574	95	1	we	we	PRON
ejpam-574	95	2	have	have	VERB
ejpam-574	95	3	x	x	NOUN
ejpam-574	95	4	–	–	PUNCT
ejpam-574	95	5	a	a	DET
ejpam-574	95	6	⊂	⊂	X
ejpam-574	95	7	x	x	X
ejpam-574	95	8	–	–	PUNCT
ejpam-574	95	9	f.	f.	PROPN
ejpam-574	95	10	by	by	ADP
ejpam-574	95	11	assumption	assumption	NOUN
ejpam-574	95	12	,	,	PUNCT
ejpam-574	95	13	cl(x	cl(x	X
ejpam-574	95	14	–	–	PUNCT
ejpam-574	95	15	a	a	X
ejpam-574	95	16	)	)	PUNCT
ejpam-574	95	17	⊂	⊂	PROPN
ejpam-574	95	18	(	(	PUNCT
ejpam-574	95	19	x	x	X
ejpam-574	95	20	–	–	PUNCT
ejpam-574	95	21	f	f	X
ejpam-574	95	22	)	)	PUNCT
ejpam-574	95	23	∪	∪	ADP
ejpam-574	95	24	u	u	NOUN
ejpam-574	95	25	,	,	PUNCT
ejpam-574	95	26	for	for	ADP
ejpam-574	95	27	some	some	DET
ejpam-574	95	28	u	u	PROPN
ejpam-574	95	29	∈	∈	PROPN
ejpam-574	95	30	i.	i.	NOUN
ejpam-574	95	31	this	this	PRON
ejpam-574	95	32	implies	imply	VERB
ejpam-574	95	33	x–((x	x–((x	NOUN
ejpam-574	95	34	–	–	PUNCT
ejpam-574	95	35	f)∪u	f)∪u	PROPN
ejpam-574	95	36	)	)	PUNCT
ejpam-574	96	1	⊂	⊂	PROPN
ejpam-574	97	1	x–(cl(x	x–(cl(x	PROPN
ejpam-574	97	2	–	–	PUNCT
ejpam-574	97	3	a	a	NOUN
ejpam-574	97	4	)	)	PUNCT
ejpam-574	97	5	)	)	PUNCT
ejpam-574	97	6	and	and	CCONJ
ejpam-574	97	7	hence	hence	ADV
ejpam-574	97	8	f	f	PROPN
ejpam-574	97	9	–	–	PUNCT
ejpam-574	97	10	u	u	X
ejpam-574	97	11	⊂	⊂	PROPN
ejpam-574	97	12	int(a	int(a	PROPN
ejpam-574	97	13	)	)	PUNCT
ejpam-574	97	14	.	.	PUNCT
ejpam-574	98	1	conversely	conversely	ADV
ejpam-574	98	2	,	,	PUNCT
ejpam-574	98	3	assume	assume	VERB
ejpam-574	98	4	that	that	SCONJ
ejpam-574	98	5	f⊂a	f⊂a	PROPN
ejpam-574	98	6	and	and	CCONJ
ejpam-574	98	7	f	f	PROPN
ejpam-574	98	8	is	be	AUX
ejpam-574	98	9	closed	close	VERB
ejpam-574	98	10	imply	imply	ADV
ejpam-574	98	11	f	f	X
ejpam-574	98	12	–	–	PUNCT
ejpam-574	98	13	u⊂int(a	u⊂int(a	NUM
ejpam-574	98	14	)	)	PUNCT
ejpam-574	98	15	,	,	PUNCT
ejpam-574	98	16	for	for	ADP
ejpam-574	98	17	some	some	DET
ejpam-574	98	18	u∈i	u∈i	NOUN
ejpam-574	98	19	.	.	PUNCT
ejpam-574	99	1	consider	consider	VERB
ejpam-574	99	2	an	an	DET
ejpam-574	99	3	open	open	ADJ
ejpam-574	99	4	set	set	NOUN
ejpam-574	99	5	g	g	PROPN
ejpam-574	99	6	such	such	ADJ
ejpam-574	99	7	that	that	SCONJ
ejpam-574	99	8	x	x	NOUN
ejpam-574	99	9	–	–	PUNCT
ejpam-574	99	10	a⊂g	a⊂g	ADJ
ejpam-574	99	11	.	.	PUNCT
ejpam-574	100	1	then	then	ADV
ejpam-574	100	2	x	x	X
ejpam-574	100	3	–	–	PUNCT
ejpam-574	100	4	g⊂a	g⊂a	NOUN
ejpam-574	100	5	.	.	PUNCT
ejpam-574	101	1	by	by	ADP
ejpam-574	101	2	assumption	assumption	NOUN
ejpam-574	101	3	,	,	PUNCT
ejpam-574	101	4	(	(	PUNCT
ejpam-574	101	5	x	x	X
ejpam-574	101	6	–	–	PUNCT
ejpam-574	101	7	g)–u⊂int(a	g)–u⊂int(a	NOUN
ejpam-574	101	8	)	)	PUNCT
ejpam-574	101	9	=	=	SYM
ejpam-574	101	10	x	x	X
ejpam-574	101	11	–	–	PUNCT
ejpam-574	101	12	cl(x	cl(x	X
ejpam-574	101	13	–	–	PUNCT
ejpam-574	101	14	a	a	PRON
ejpam-574	101	15	)	)	PUNCT
ejpam-574	101	16	.	.	PUNCT
ejpam-574	102	1	this	this	PRON
ejpam-574	102	2	gives	give	VERB
ejpam-574	102	3	that	that	DET
ejpam-574	102	4	x–(g∪u)⊂x	x–(g∪u)⊂x	NOUN
ejpam-574	102	5	–	–	PUNCT
ejpam-574	102	6	cl(x	cl(x	X
ejpam-574	102	7	–	–	PUNCT
ejpam-574	102	8	a	a	PRON
ejpam-574	102	9	)	)	PUNCT
ejpam-574	102	10	.	.	PUNCT
ejpam-574	103	1	then	then	ADV
ejpam-574	103	2	,	,	PUNCT
ejpam-574	103	3	cl(x	cl(x	X
ejpam-574	103	4	–	–	PUNCT
ejpam-574	103	5	a)⊂g∪u	a)⊂g∪u	NOUN
ejpam-574	103	6	,	,	PUNCT
ejpam-574	103	7	for	for	ADP
ejpam-574	103	8	some	some	DET
ejpam-574	103	9	u∈i	u∈i	NOUN
ejpam-574	103	10	.	.	PUNCT
ejpam-574	104	1	this	this	PRON
ejpam-574	104	2	shows	show	VERB
ejpam-574	104	3	that	that	SCONJ
ejpam-574	104	4	cl(x	cl(x	NOUN
ejpam-574	104	5	–	–	PUNCT
ejpam-574	104	6	a)–g∈i	a)–g∈i	NOUN
ejpam-574	104	7	.	.	PUNCT
ejpam-574	105	1	hence	hence	ADV
ejpam-574	105	2	x	x	X
ejpam-574	105	3	–	–	PUNCT
ejpam-574	105	4	a	a	PRON
ejpam-574	105	5	is	be	AUX
ejpam-574	105	6	ig	ig	PRON
ejpam-574	105	7	-	-	ADJ
ejpam-574	105	8	closed	closed	ADJ
ejpam-574	105	9	.	.	PUNCT
ejpam-574	106	1	recall	recall	VERB
ejpam-574	106	2	that	that	SCONJ
ejpam-574	106	3	the	the	DET
ejpam-574	106	4	sets	set	NOUN
ejpam-574	106	5	a	a	PRON
ejpam-574	106	6	and	and	CCONJ
ejpam-574	106	7	b	b	NOUN
ejpam-574	106	8	are	be	AUX
ejpam-574	106	9	said	say	VERB
ejpam-574	106	10	to	to	PART
ejpam-574	106	11	be	be	AUX
ejpam-574	106	12	separated	separate	VERB
ejpam-574	106	13	if	if	SCONJ
ejpam-574	106	14	cl(a)∩b	cl(a)∩b	PROPN
ejpam-574	106	15	=	=	SYM
ejpam-574	106	16	∅	∅	NOUN
ejpam-574	106	17	and	and	CCONJ
ejpam-574	106	18	a	a	DET
ejpam-574	106	19	∩	∩	ADJ
ejpam-574	106	20	cl(b	cl(b	NOUN
ejpam-574	106	21	)	)	PUNCT
ejpam-574	106	22	=	=	PUNCT
ejpam-574	106	23	∅.	∅.	NOUN
ejpam-574	106	24	theorem	theorem	VERB
ejpam-574	106	25	7	7	NUM
ejpam-574	106	26	.	.	PUNCT
ejpam-574	107	1	if	if	SCONJ
ejpam-574	107	2	a	a	PRON
ejpam-574	107	3	and	and	CCONJ
ejpam-574	107	4	b	b	NOUN
ejpam-574	107	5	are	be	AUX
ejpam-574	107	6	separated	separate	VERB
ejpam-574	107	7	ig	ig	ADJ
ejpam-574	107	8	-	-	ADJ
ejpam-574	107	9	open	open	ADJ
ejpam-574	107	10	sets	set	NOUN
ejpam-574	107	11	in	in	ADP
ejpam-574	107	12	(	(	PUNCT
ejpam-574	107	13	x	x	INTJ
ejpam-574	107	14	,	,	PUNCT
ejpam-574	107	15	τ	τ	PROPN
ejpam-574	107	16	)	)	PUNCT
ejpam-574	107	17	,	,	PUNCT
ejpam-574	107	18	then	then	ADV
ejpam-574	107	19	a	a	DET
ejpam-574	107	20	∪	∪	X
ejpam-574	107	21	b	b	NOUN
ejpam-574	107	22	is	be	AUX
ejpam-574	107	23	ig	ig	NOUN
ejpam-574	107	24	-	-	ADJ
ejpam-574	107	25	open	open	ADJ
ejpam-574	107	26	.	.	PUNCT
ejpam-574	108	1	proof	proof	NOUN
ejpam-574	108	2	.	.	PUNCT
ejpam-574	109	1	suppose	suppose	VERB
ejpam-574	109	2	a	a	PRON
ejpam-574	109	3	and	and	CCONJ
ejpam-574	109	4	b	b	NOUN
ejpam-574	109	5	are	be	AUX
ejpam-574	109	6	separated	separate	VERB
ejpam-574	109	7	ig	ig	ADJ
ejpam-574	109	8	-	-	ADJ
ejpam-574	109	9	open	open	ADJ
ejpam-574	109	10	sets	set	NOUN
ejpam-574	109	11	in	in	ADP
ejpam-574	109	12	(	(	PUNCT
ejpam-574	109	13	x	x	INTJ
ejpam-574	109	14	,	,	PUNCT
ejpam-574	109	15	τ	τ	PROPN
ejpam-574	109	16	)	)	PUNCT
ejpam-574	109	17	and	and	CCONJ
ejpam-574	109	18	f	f	PROPN
ejpam-574	109	19	be	be	AUX
ejpam-574	109	20	a	a	DET
ejpam-574	109	21	closed	closed	ADJ
ejpam-574	109	22	subset	subset	NOUN
ejpam-574	109	23	of	of	ADP
ejpam-574	109	24	a∪b	a∪b	NOUN
ejpam-574	109	25	.	.	PUNCT
ejpam-574	110	1	then	then	ADV
ejpam-574	110	2	f∩cl(a)⊂a	f∩cl(a)⊂a	PROPN
ejpam-574	110	3	and	and	CCONJ
ejpam-574	110	4	f∩cl(b)⊂b	f∩cl(b)⊂b	PROPN
ejpam-574	110	5	.	.	PUNCT
ejpam-574	110	6	by	by	ADP
ejpam-574	110	7	assumption	assumption	NOUN
ejpam-574	110	8	,	,	PUNCT
ejpam-574	110	9	(	(	PUNCT
ejpam-574	110	10	f∩cl(a))–u1	f∩cl(a))–u1	NOUN
ejpam-574	110	11	⊂	⊂	PROPN
ejpam-574	110	12	int(a	int(a	PROPN
ejpam-574	110	13	)	)	PUNCT
ejpam-574	110	14	and	and	CCONJ
ejpam-574	110	15	(	(	PUNCT
ejpam-574	110	16	f∩cl(b))–u2	f∩cl(b))–u2	X
ejpam-574	110	17	⊂	⊂	X
ejpam-574	110	18	int(b	int(b	PROPN
ejpam-574	110	19	)	)	PUNCT
ejpam-574	110	20	,	,	PUNCT
ejpam-574	110	21	for	for	ADP
ejpam-574	110	22	some	some	DET
ejpam-574	110	23	u1	u1	NOUN
ejpam-574	110	24	,	,	PUNCT
ejpam-574	110	25	u2	u2	PROPN
ejpam-574	110	26	∈	∈	PROPN
ejpam-574	110	27	i.	i.	NOUN
ejpam-574	110	28	this	this	PRON
ejpam-574	110	29	mean	mean	VERB
ejpam-574	110	30	that	that	SCONJ
ejpam-574	110	31	(	(	PUNCT
ejpam-574	110	32	(	(	PUNCT
ejpam-574	110	33	f∩cl(a))–int(a	f∩cl(a))–int(a	PROPN
ejpam-574	110	34	)	)	PUNCT
ejpam-574	110	35	)	)	PUNCT
ejpam-574	111	1	∈	∈	PROPN
ejpam-574	112	1	i	i	PRON
ejpam-574	112	2	and	and	CCONJ
ejpam-574	112	3	(	(	PUNCT
ejpam-574	112	4	f∩cl(b))–int(b)∈i	f∩cl(b))–int(b)∈i	ADJ
ejpam-574	112	5	.	.	PUNCT
ejpam-574	113	1	then	then	ADV
ejpam-574	113	2	(	(	PUNCT
ejpam-574	113	3	(	(	PUNCT
ejpam-574	113	4	f∩cl(a))–int(a))∪((f∩cl(b))–int(b))∈i	f∩cl(a))–int(a))∪((f∩cl(b))–int(b))∈i	NOUN
ejpam-574	113	5	.	.	PUNCT
ejpam-574	114	1	hence	hence	ADV
ejpam-574	114	2	(	(	PUNCT
ejpam-574	114	3	f∩(cl(a)∪cl(b))–(int(a)∪int(b	f∩(cl(a)∪cl(b))–(int(a)∪int(b	NOUN
ejpam-574	114	4	)	)	PUNCT
ejpam-574	114	5	)	)	PUNCT
ejpam-574	114	6	)	)	PUNCT
ejpam-574	115	1	∈	∈	PROPN
ejpam-574	115	2	i.	i.	NOUN
ejpam-574	115	3	but	but	CCONJ
ejpam-574	115	4	f	f	X
ejpam-574	115	5	=	=	NOUN
ejpam-574	115	6	f∩(a∪b	f∩(a∪b	X
ejpam-574	115	7	)	)	PUNCT
ejpam-574	115	8	⊂	⊂	NOUN
ejpam-574	115	9	f∩cl(a∪b	f∩cl(a∪b	NUM
ejpam-574	115	10	)	)	PUNCT
ejpam-574	115	11	,	,	PUNCT
ejpam-574	115	12	and	and	CCONJ
ejpam-574	115	13	we	we	PRON
ejpam-574	115	14	have	have	VERB
ejpam-574	115	15	f	f	X
ejpam-574	115	16	−	−	PROPN
ejpam-574	115	17	int(a∪	int(a∪	NOUN
ejpam-574	115	18	b	b	NOUN
ejpam-574	115	19	)	)	PUNCT
ejpam-574	115	20	⊂	⊂	PROPN
ejpam-574	115	21	(	(	PUNCT
ejpam-574	115	22	f	f	PROPN
ejpam-574	115	23	∩	∩	PROPN
ejpam-574	115	24	cl(a∪	cl(a∪	PROPN
ejpam-574	116	1	b))−	b))−	AUX
ejpam-574	116	2	int(a∪	int(a∪	NOUN
ejpam-574	116	3	b	b	X
ejpam-574	116	4	)	)	PUNCT
ejpam-574	117	1	⊂	⊂	PROPN
ejpam-574	117	2	(	(	PUNCT
ejpam-574	117	3	f	f	PROPN
ejpam-574	117	4	∩	∩	NOUN
ejpam-574	117	5	cl(a∪	cl(a∪	PROPN
ejpam-574	117	6	b))−	b))−	PROPN
ejpam-574	117	7	(	(	PUNCT
ejpam-574	117	8	int(a)∪	int(a)∪	PROPN
ejpam-574	117	9	int(b	int(b	PROPN
ejpam-574	117	10	)	)	PUNCT
ejpam-574	117	11	)	)	PUNCT
ejpam-574	118	1	∈	∈	PROPN
ejpam-574	119	1	i	i	PRON
ejpam-574	119	2	hence	hence	ADV
ejpam-574	119	3	,	,	PUNCT
ejpam-574	119	4	f	f	X
ejpam-574	119	5	–	–	PUNCT
ejpam-574	119	6	u⊂int(a∪b	u⊂int(a∪b	ADJ
ejpam-574	119	7	)	)	PUNCT
ejpam-574	119	8	,	,	PUNCT
ejpam-574	119	9	for	for	ADP
ejpam-574	119	10	some	some	DET
ejpam-574	119	11	u∈i	u∈i	NOUN
ejpam-574	119	12	.	.	PUNCT
ejpam-574	120	1	this	this	PRON
ejpam-574	120	2	proves	prove	VERB
ejpam-574	120	3	that	that	SCONJ
ejpam-574	120	4	a∪b	a∪b	NOUN
ejpam-574	120	5	is	be	AUX
ejpam-574	120	6	ig	ig	NOUN
ejpam-574	120	7	-	-	ADJ
ejpam-574	120	8	open	open	ADJ
ejpam-574	120	9	.	.	PUNCT
ejpam-574	121	1	corollary	corollary	ADJ
ejpam-574	121	2	1	1	NUM
ejpam-574	121	3	.	.	PUNCT
ejpam-574	122	1	let	let	VERB
ejpam-574	122	2	a	a	PRON
ejpam-574	122	3	and	and	CCONJ
ejpam-574	122	4	b	b	NOUN
ejpam-574	122	5	are	be	AUX
ejpam-574	122	6	ig	ig	PRON
ejpam-574	122	7	-	-	ADJ
ejpam-574	122	8	closed	closed	ADJ
ejpam-574	122	9	sets	set	NOUN
ejpam-574	122	10	and	and	CCONJ
ejpam-574	122	11	suppose	suppose	VERB
ejpam-574	122	12	x	x	X
ejpam-574	122	13	–	–	PUNCT
ejpam-574	122	14	a	a	PRON
ejpam-574	122	15	and	and	CCONJ
ejpam-574	122	16	x	x	X
ejpam-574	122	17	–	–	PUNCT
ejpam-574	122	18	b	b	NOUN
ejpam-574	122	19	are	be	AUX
ejpam-574	122	20	separated	separate	VERB
ejpam-574	122	21	in	in	ADP
ejpam-574	122	22	(	(	PUNCT
ejpam-574	122	23	x	x	INTJ
ejpam-574	122	24	,	,	PUNCT
ejpam-574	122	25	τ	τ	PROPN
ejpam-574	122	26	)	)	PUNCT
ejpam-574	122	27	.	.	PUNCT
ejpam-574	123	1	then	then	ADV
ejpam-574	123	2	a∩b	a∩b	PROPN
ejpam-574	123	3	is	be	AUX
ejpam-574	123	4	ig	ig	PRON
ejpam-574	123	5	-	-	ADJ
ejpam-574	123	6	closed	closed	ADJ
ejpam-574	123	7	.	.	PUNCT
ejpam-574	124	1	corollary	corollary	ADJ
ejpam-574	124	2	2	2	NUM
ejpam-574	124	3	.	.	PUNCT
ejpam-574	125	1	if	if	SCONJ
ejpam-574	125	2	a	a	PRON
ejpam-574	125	3	and	and	CCONJ
ejpam-574	125	4	b	b	NOUN
ejpam-574	125	5	are	be	AUX
ejpam-574	125	6	ig	ig	ADJ
ejpam-574	125	7	-	-	ADJ
ejpam-574	125	8	open	open	ADJ
ejpam-574	125	9	sets	set	NOUN
ejpam-574	125	10	in	in	ADP
ejpam-574	125	11	(	(	PUNCT
ejpam-574	125	12	x	x	INTJ
ejpam-574	125	13	,	,	PUNCT
ejpam-574	125	14	τ	τ	PROPN
ejpam-574	125	15	)	)	PUNCT
ejpam-574	125	16	,	,	PUNCT
ejpam-574	125	17	then	then	ADV
ejpam-574	125	18	a∩b	a∩b	PROPN
ejpam-574	125	19	is	be	AUX
ejpam-574	125	20	ig	ig	NOUN
ejpam-574	125	21	-	-	ADJ
ejpam-574	125	22	open	open	ADJ
ejpam-574	125	23	.	.	PUNCT
ejpam-574	126	1	proof	proof	NOUN
ejpam-574	126	2	.	.	PUNCT
ejpam-574	127	1	if	if	SCONJ
ejpam-574	127	2	a	a	PRON
ejpam-574	127	3	and	and	CCONJ
ejpam-574	127	4	b	b	NOUN
ejpam-574	127	5	are	be	AUX
ejpam-574	127	6	ig	ig	PRON
ejpam-574	127	7	-	-	ADJ
ejpam-574	127	8	open	open	ADJ
ejpam-574	127	9	,	,	PUNCT
ejpam-574	127	10	then	then	ADV
ejpam-574	127	11	x	x	X
ejpam-574	127	12	–	–	PUNCT
ejpam-574	127	13	a	a	PRON
ejpam-574	127	14	and	and	CCONJ
ejpam-574	127	15	x	x	X
ejpam-574	127	16	–	–	PUNCT
ejpam-574	127	17	b	b	NUM
ejpam-574	127	18	are	be	AUX
ejpam-574	127	19	ig	ig	PRON
ejpam-574	127	20	-	-	VERB
ejpam-574	127	21	closed	closed	ADJ
ejpam-574	127	22	.	.	PUNCT
ejpam-574	128	1	by	by	ADP
ejpam-574	128	2	theorem	theorem	NOUN
ejpam-574	128	3	2	2	NUM
ejpam-574	128	4	,	,	PUNCT
ejpam-574	128	5	x–(a∩b	x–(a∩b	PROPN
ejpam-574	128	6	)	)	PUNCT
ejpam-574	128	7	is	be	AUX
ejpam-574	128	8	ig	ig	PRON
ejpam-574	128	9	-	-	ADJ
ejpam-574	128	10	closed	closed	ADJ
ejpam-574	128	11	,	,	PUNCT
ejpam-574	128	12	which	which	PRON
ejpam-574	128	13	implies	imply	VERB
ejpam-574	128	14	a∩b	a∩b	PROPN
ejpam-574	128	15	is	be	AUX
ejpam-574	128	16	ig	ig	NOUN
ejpam-574	128	17	-	-	ADJ
ejpam-574	128	18	open	open	ADJ
ejpam-574	128	19	.	.	PUNCT
ejpam-574	129	1	references	reference	NOUN
ejpam-574	129	2	150	150	NUM
ejpam-574	129	3	theorem	theorem	NOUN
ejpam-574	129	4	8	8	NUM
ejpam-574	129	5	.	.	PUNCT
ejpam-574	130	1	if	if	SCONJ
ejpam-574	130	2	a⊂b⊂x	a⊂b⊂x	PROPN
ejpam-574	130	3	,	,	PUNCT
ejpam-574	130	4	a	a	PRON
ejpam-574	130	5	is	be	AUX
ejpam-574	130	6	ig	ig	NOUN
ejpam-574	130	7	-	-	ADJ
ejpam-574	130	8	open	open	ADJ
ejpam-574	130	9	relative	relative	ADJ
ejpam-574	130	10	to	to	ADP
ejpam-574	130	11	b	b	NOUN
ejpam-574	130	12	and	and	CCONJ
ejpam-574	130	13	b	b	PROPN
ejpam-574	130	14	is	be	AUX
ejpam-574	130	15	ig	ig	NOUN
ejpam-574	130	16	-	-	ADJ
ejpam-574	130	17	open	open	ADJ
ejpam-574	130	18	relative	relative	ADJ
ejpam-574	130	19	to	to	ADP
ejpam-574	130	20	x	x	PRON
ejpam-574	130	21	,	,	PUNCT
ejpam-574	130	22	then	then	ADV
ejpam-574	130	23	a	a	PRON
ejpam-574	130	24	is	be	AUX
ejpam-574	130	25	ig	ig	NOUN
ejpam-574	130	26	-	-	ADJ
ejpam-574	130	27	open	open	ADJ
ejpam-574	130	28	relative	relative	ADJ
ejpam-574	130	29	to	to	ADP
ejpam-574	130	30	x.	x.	NOUN
ejpam-574	130	31	proof	proof	NOUN
ejpam-574	130	32	.	.	PUNCT
ejpam-574	131	1	suppose	suppose	VERB
ejpam-574	131	2	a⊂b⊂x	a⊂b⊂x	PROPN
ejpam-574	131	3	,	,	PUNCT
ejpam-574	131	4	a	a	PRON
ejpam-574	131	5	is	be	AUX
ejpam-574	131	6	ig	ig	NOUN
ejpam-574	131	7	-	-	ADJ
ejpam-574	131	8	open	open	ADJ
ejpam-574	131	9	relative	relative	ADJ
ejpam-574	131	10	to	to	ADP
ejpam-574	131	11	b	b	NOUN
ejpam-574	131	12	and	and	CCONJ
ejpam-574	131	13	b	b	PROPN
ejpam-574	131	14	is	be	AUX
ejpam-574	131	15	ig	ig	NOUN
ejpam-574	131	16	-	-	ADJ
ejpam-574	131	17	open	open	ADJ
ejpam-574	131	18	relative	relative	ADJ
ejpam-574	131	19	to	to	ADP
ejpam-574	131	20	x.	x.	NOUN
ejpam-574	131	21	suppose	suppose	VERB
ejpam-574	131	22	f⊂a	f⊂a	PROPN
ejpam-574	131	23	and	and	CCONJ
ejpam-574	131	24	f	f	PROPN
ejpam-574	131	25	is	be	AUX
ejpam-574	131	26	closed	closed	ADJ
ejpam-574	131	27	.	.	PUNCT
ejpam-574	132	1	since	since	SCONJ
ejpam-574	132	2	a	a	PRON
ejpam-574	132	3	is	be	AUX
ejpam-574	132	4	ig	ig	NOUN
ejpam-574	132	5	-	-	ADJ
ejpam-574	132	6	open	open	ADJ
ejpam-574	132	7	relative	relative	ADJ
ejpam-574	132	8	to	to	ADP
ejpam-574	132	9	b	b	NUM
ejpam-574	132	10	,	,	PUNCT
ejpam-574	132	11	by	by	ADP
ejpam-574	132	12	theorem	theorem	NOUN
ejpam-574	132	13	6	6	NUM
ejpam-574	132	14	,	,	PUNCT
ejpam-574	132	15	f	f	X
ejpam-574	132	16	–	–	PUNCT
ejpam-574	132	17	u1	u1	NOUN
ejpam-574	132	18	⊂	⊂	PROPN
ejpam-574	132	19	intb(a	intb(a	PROPN
ejpam-574	132	20	)	)	PUNCT
ejpam-574	132	21	,	,	PUNCT
ejpam-574	132	22	for	for	ADP
ejpam-574	132	23	some	some	DET
ejpam-574	132	24	u1	u1	NOUN
ejpam-574	132	25	∈	∈	PROPN
ejpam-574	132	26	i.	i.	NOUN
ejpam-574	132	27	this	this	PRON
ejpam-574	132	28	implies	imply	VERB
ejpam-574	132	29	there	there	PRON
ejpam-574	132	30	exists	exist	VERB
ejpam-574	132	31	an	an	DET
ejpam-574	132	32	open	open	ADJ
ejpam-574	132	33	set	set	NOUN
ejpam-574	132	34	g1	g1	NOUN
ejpam-574	132	35	such	such	ADJ
ejpam-574	132	36	that	that	SCONJ
ejpam-574	132	37	f	f	X
ejpam-574	132	38	–	–	PUNCT
ejpam-574	132	39	u1	u1	NOUN
ejpam-574	132	40	⊂	⊂	PROPN
ejpam-574	132	41	g1	g1	PROPN
ejpam-574	132	42	∩	∩	ADJ
ejpam-574	132	43	b⊂a	b⊂a	NOUN
ejpam-574	132	44	,	,	PUNCT
ejpam-574	132	45	for	for	ADP
ejpam-574	132	46	some	some	DET
ejpam-574	132	47	u1	u1	NOUN
ejpam-574	132	48	∈	∈	PROPN
ejpam-574	132	49	i.	i.	NOUN
ejpam-574	132	50	since	since	SCONJ
ejpam-574	132	51	b	b	PROPN
ejpam-574	132	52	is	be	AUX
ejpam-574	132	53	ig	ig	NOUN
ejpam-574	132	54	-	-	ADJ
ejpam-574	132	55	open	open	ADJ
ejpam-574	132	56	,	,	PUNCT
ejpam-574	132	57	f	f	PROPN
ejpam-574	132	58	⊂	⊂	PROPN
ejpam-574	132	59	b	b	PROPN
ejpam-574	132	60	and	and	CCONJ
ejpam-574	132	61	f	f	PROPN
ejpam-574	132	62	is	be	AUX
ejpam-574	132	63	closed	closed	ADJ
ejpam-574	132	64	;	;	PUNCT
ejpam-574	132	65	we	we	PRON
ejpam-574	132	66	have	have	VERB
ejpam-574	132	67	f	f	PROPN
ejpam-574	132	68	–	–	PUNCT
ejpam-574	132	69	u2	u2	PROPN
ejpam-574	132	70	⊂	⊂	PROPN
ejpam-574	132	71	int(b	int(b	PROPN
ejpam-574	132	72	)	)	PUNCT
ejpam-574	132	73	,	,	PUNCT
ejpam-574	132	74	for	for	ADP
ejpam-574	132	75	some	some	DET
ejpam-574	132	76	u2	u2	PROPN
ejpam-574	132	77	∈	∈	PROPN
ejpam-574	132	78	i.	i.	NOUN
ejpam-574	132	79	this	this	PRON
ejpam-574	132	80	implies	imply	VERB
ejpam-574	132	81	there	there	PRON
ejpam-574	132	82	exists	exist	VERB
ejpam-574	132	83	an	an	DET
ejpam-574	132	84	open	open	ADJ
ejpam-574	132	85	set	set	VERB
ejpam-574	132	86	g2	g2	PROPN
ejpam-574	132	87	such	such	ADJ
ejpam-574	132	88	that	that	SCONJ
ejpam-574	132	89	f	f	PROPN
ejpam-574	132	90	–	–	PUNCT
ejpam-574	132	91	u2	u2	PROPN
ejpam-574	132	92	⊂	⊂	PROPN
ejpam-574	132	93	g2	g2	PROPN
ejpam-574	132	94	⊂	⊂	PROPN
ejpam-574	132	95	b	b	PROPN
ejpam-574	132	96	,	,	PUNCT
ejpam-574	132	97	for	for	ADP
ejpam-574	132	98	some	some	DET
ejpam-574	132	99	u2	u2	PROPN
ejpam-574	132	100	∈	∈	PROPN
ejpam-574	132	101	i.	i.	NOUN
ejpam-574	133	1	now	now	ADV
ejpam-574	133	2	f	f	X
ejpam-574	133	3	–	–	PUNCT
ejpam-574	133	4	(	(	PUNCT
ejpam-574	133	5	u1	u1	PROPN
ejpam-574	133	6	∪	∪	NOUN
ejpam-574	133	7	u2	u2	PROPN
ejpam-574	133	8	)	)	PUNCT
ejpam-574	134	1	⊂	⊂	PROPN
ejpam-574	134	2	(	(	PUNCT
ejpam-574	134	3	f	f	X
ejpam-574	134	4	–	–	PUNCT
ejpam-574	134	5	u1	u1	NOUN
ejpam-574	134	6	)	)	PUNCT
ejpam-574	134	7	∩	∩	NOUN
ejpam-574	134	8	(	(	PUNCT
ejpam-574	134	9	f	f	X
ejpam-574	134	10	–	–	PUNCT
ejpam-574	134	11	u2	u2	NOUN
ejpam-574	134	12	)	)	PUNCT
ejpam-574	134	13	⊂	⊂	PROPN
ejpam-574	134	14	g1	g1	PROPN
ejpam-574	134	15	∩	∩	PROPN
ejpam-574	134	16	g2	g2	PROPN
ejpam-574	134	17	⊂	⊂	PROPN
ejpam-574	134	18	g1	g1	PROPN
ejpam-574	134	19	∩	∩	PROPN
ejpam-574	134	20	b	b	PROPN
ejpam-574	134	21	⊂	⊂	PROPN
ejpam-574	134	22	a.	a.	NOUN
ejpam-574	135	1	this	this	PRON
ejpam-574	135	2	implies	imply	VERB
ejpam-574	135	3	that	that	SCONJ
ejpam-574	135	4	f	f	PROPN
ejpam-574	135	5	–	–	PUNCT
ejpam-574	135	6	(	(	PUNCT
ejpam-574	135	7	u1	u1	PROPN
ejpam-574	135	8	∪	∪	NOUN
ejpam-574	135	9	u2	u2	PROPN
ejpam-574	135	10	)	)	PUNCT
ejpam-574	135	11	⊂	⊂	PROPN
ejpam-574	136	1	int(a	int(a	PROPN
ejpam-574	136	2	)	)	PUNCT
ejpam-574	136	3	,	,	PUNCT
ejpam-574	136	4	for	for	SCONJ
ejpam-574	136	5	some	some	DET
ejpam-574	136	6	u1	u1	NOUN
ejpam-574	136	7	∪	∪	ADP
ejpam-574	136	8	u2	u2	PROPN
ejpam-574	136	9	∈	∈	PROPN
ejpam-574	137	1	i	i	PRON
ejpam-574	137	2	and	and	CCONJ
ejpam-574	137	3	hence	hence	ADV
ejpam-574	137	4	a	a	PRON
ejpam-574	137	5	is	be	AUX
ejpam-574	137	6	ig	ig	NOUN
ejpam-574	137	7	-	-	ADJ
ejpam-574	137	8	open	open	ADJ
ejpam-574	137	9	relative	relative	ADJ
ejpam-574	137	10	to	to	ADP
ejpam-574	137	11	x.	x.	NOUN
ejpam-574	137	12	theorem	theorem	VERB
ejpam-574	137	13	9	9	NUM
ejpam-574	137	14	.	.	PUNCT
ejpam-574	138	1	if	if	SCONJ
ejpam-574	138	2	int(a	int(a	PROPN
ejpam-574	138	3	)	)	PUNCT
ejpam-574	139	1	⊂	⊂	PROPN
ejpam-574	140	1	b	b	X
ejpam-574	140	2	⊂	⊂	PROPN
ejpam-574	140	3	a	a	PROPN
ejpam-574	141	1	and	and	CCONJ
ejpam-574	141	2	if	if	SCONJ
ejpam-574	141	3	a	a	PRON
ejpam-574	141	4	is	be	AUX
ejpam-574	141	5	ig	ig	NOUN
ejpam-574	141	6	-	-	ADJ
ejpam-574	141	7	open	open	ADJ
ejpam-574	141	8	in	in	ADP
ejpam-574	141	9	(	(	PUNCT
ejpam-574	141	10	x	x	INTJ
ejpam-574	141	11	,	,	PUNCT
ejpam-574	141	12	τ	τ	PROPN
ejpam-574	141	13	)	)	PUNCT
ejpam-574	141	14	,	,	PUNCT
ejpam-574	141	15	then	then	ADV
ejpam-574	141	16	b	b	PROPN
ejpam-574	141	17	is	be	AUX
ejpam-574	141	18	ig	ig	NOUN
ejpam-574	141	19	-	-	ADJ
ejpam-574	141	20	open	open	ADJ
ejpam-574	141	21	in	in	ADP
ejpam-574	141	22	x.	x.	NOUN
ejpam-574	141	23	proof	proof	NOUN
ejpam-574	141	24	.	.	PUNCT
ejpam-574	142	1	suppose	suppose	VERB
ejpam-574	142	2	int(a	int(a	X
ejpam-574	142	3	)	)	PUNCT
ejpam-574	142	4	⊂	⊂	PROPN
ejpam-574	143	1	b	b	X
ejpam-574	143	2	⊂	⊂	PROPN
ejpam-574	143	3	a	a	PROPN
ejpam-574	143	4	and	and	CCONJ
ejpam-574	143	5	a	a	PRON
ejpam-574	143	6	is	be	AUX
ejpam-574	143	7	ig	ig	NOUN
ejpam-574	143	8	-	-	ADJ
ejpam-574	143	9	open	open	ADJ
ejpam-574	143	10	.	.	PUNCT
ejpam-574	144	1	then	then	ADV
ejpam-574	144	2	x	x	X
ejpam-574	144	3	–	–	PUNCT
ejpam-574	144	4	a	a	PRON
ejpam-574	144	5	⊂	⊂	X
ejpam-574	144	6	x	x	PROPN
ejpam-574	144	7	–	–	PUNCT
ejpam-574	144	8	b	b	X
ejpam-574	144	9	⊂	⊂	ADJ
ejpam-574	144	10	cl(x	cl(x	X
ejpam-574	144	11	–	–	PUNCT
ejpam-574	144	12	a	a	NOUN
ejpam-574	144	13	)	)	PUNCT
ejpam-574	144	14	and	and	CCONJ
ejpam-574	144	15	x	x	X
ejpam-574	144	16	–	–	PUNCT
ejpam-574	144	17	a	a	PRON
ejpam-574	144	18	is	be	AUX
ejpam-574	144	19	ig	ig	PRON
ejpam-574	144	20	-	-	ADJ
ejpam-574	144	21	closed	closed	ADJ
ejpam-574	144	22	.	.	PUNCT
ejpam-574	145	1	by	by	ADP
ejpam-574	145	2	theorem	theorem	NOUN
ejpam-574	145	3	3	3	NUM
ejpam-574	145	4	,	,	PUNCT
ejpam-574	145	5	x	x	X
ejpam-574	145	6	–	–	PUNCT
ejpam-574	145	7	b	b	PROPN
ejpam-574	145	8	is	be	AUX
ejpam-574	145	9	ig	ig	PRON
ejpam-574	145	10	-	-	PUNCT
ejpam-574	145	11	closed	closed	ADJ
ejpam-574	145	12	and	and	CCONJ
ejpam-574	145	13	hence	hence	ADV
ejpam-574	145	14	b	b	PROPN
ejpam-574	145	15	is	be	AUX
ejpam-574	145	16	ig	ig	PRON
ejpam-574	145	17	-	-	ADJ
ejpam-574	145	18	open	open	ADJ
ejpam-574	145	19	.	.	PUNCT
ejpam-574	146	1	theorem	theorem	ADJ
ejpam-574	146	2	10	10	NUM
ejpam-574	146	3	.	.	PUNCT
ejpam-574	147	1	a	a	DET
ejpam-574	147	2	set	set	NOUN
ejpam-574	147	3	a	a	PRON
ejpam-574	147	4	is	be	AUX
ejpam-574	147	5	ig	ig	NOUN
ejpam-574	147	6	-	-	VERB
ejpam-574	147	7	closed	closed	ADJ
ejpam-574	147	8	in	in	ADP
ejpam-574	147	9	(	(	PUNCT
ejpam-574	147	10	x	x	INTJ
ejpam-574	147	11	,	,	PUNCT
ejpam-574	147	12	τ	τ	X
ejpam-574	147	13	)	)	PUNCT
ejpam-574	147	14	if	if	SCONJ
ejpam-574	147	15	and	and	CCONJ
ejpam-574	147	16	only	only	ADV
ejpam-574	147	17	if	if	SCONJ
ejpam-574	147	18	cl(a)–a	cl(a)–a	PROPN
ejpam-574	147	19	is	be	AUX
ejpam-574	147	20	ig	ig	NOUN
ejpam-574	147	21	-	-	ADJ
ejpam-574	147	22	open	open	ADJ
ejpam-574	147	23	.	.	PUNCT
ejpam-574	148	1	proof	proof	NOUN
ejpam-574	148	2	.	.	PUNCT
ejpam-574	149	1	necessity	necessity	NOUN
ejpam-574	149	2	:	:	PUNCT
ejpam-574	149	3	suppose	suppose	VERB
ejpam-574	149	4	f	f	PROPN
ejpam-574	149	5	⊂	⊂	PROPN
ejpam-574	149	6	cl(a)–a	cl(a)–a	PROPN
ejpam-574	149	7	and	and	CCONJ
ejpam-574	149	8	f	f	PROPN
ejpam-574	149	9	be	be	AUX
ejpam-574	149	10	closed	close	VERB
ejpam-574	149	11	.	.	PUNCT
ejpam-574	150	1	then	then	ADV
ejpam-574	150	2	f	f	PROPN
ejpam-574	150	3	∈	∈	PROPN
ejpam-574	150	4	i.	i.	NOUN
ejpam-574	150	5	this	this	PRON
ejpam-574	150	6	implies	imply	VERB
ejpam-574	150	7	that	that	SCONJ
ejpam-574	150	8	f	f	X
ejpam-574	150	9	–	–	PUNCT
ejpam-574	150	10	u	u	NOUN
ejpam-574	150	11	=	=	NOUN
ejpam-574	150	12	∅	∅	NOUN
ejpam-574	150	13	,	,	PUNCT
ejpam-574	150	14	for	for	ADP
ejpam-574	150	15	some	some	DET
ejpam-574	150	16	u	u	PROPN
ejpam-574	150	17	∈	∈	PROPN
ejpam-574	150	18	i.	i.	NOUN
ejpam-574	150	19	clearly	clearly	ADV
ejpam-574	150	20	,	,	PUNCT
ejpam-574	150	21	f	f	X
ejpam-574	150	22	–	–	PUNCT
ejpam-574	150	23	u	u	PROPN
ejpam-574	150	24	⊂	⊂	PROPN
ejpam-574	150	25	int(cl(a)–a	int(cl(a)–a	PROPN
ejpam-574	150	26	)	)	PUNCT
ejpam-574	150	27	.	.	PUNCT
ejpam-574	151	1	by	by	ADP
ejpam-574	151	2	theorem	theorem	NOUN
ejpam-574	151	3	6	6	NUM
ejpam-574	151	4	cl(a)–a	cl(a)–a	PROPN
ejpam-574	151	5	is	be	AUX
ejpam-574	151	6	ig	ig	NOUN
ejpam-574	151	7	-	-	ADJ
ejpam-574	151	8	open	open	ADJ
ejpam-574	151	9	.	.	PUNCT
ejpam-574	152	1	sufficiency	sufficiency	NOUN
ejpam-574	152	2	:	:	PUNCT
ejpam-574	152	3	suppose	suppose	VERB
ejpam-574	152	4	a	a	DET
ejpam-574	152	5	⊂	⊂	PROPN
ejpam-574	152	6	g	g	PROPN
ejpam-574	152	7	and	and	CCONJ
ejpam-574	152	8	g	g	PROPN
ejpam-574	152	9	is	be	AUX
ejpam-574	152	10	open	open	ADJ
ejpam-574	152	11	in	in	ADP
ejpam-574	152	12	(	(	PUNCT
ejpam-574	152	13	x	x	INTJ
ejpam-574	152	14	,	,	PUNCT
ejpam-574	152	15	τ	τ	PROPN
ejpam-574	152	16	)	)	PUNCT
ejpam-574	152	17	.	.	PUNCT
ejpam-574	153	1	then	then	ADV
ejpam-574	153	2	cl(a)∩(x	cl(a)∩(x	NOUN
ejpam-574	153	3	–	–	PUNCT
ejpam-574	153	4	g	g	NOUN
ejpam-574	153	5	)	)	PUNCT
ejpam-574	153	6	⊂	⊂	PROPN
ejpam-574	153	7	cl(a	cl(a	X
ejpam-574	153	8	)	)	PUNCT
ejpam-574	153	9	∩	∩	NOUN
ejpam-574	153	10	(	(	PUNCT
ejpam-574	153	11	x	x	X
ejpam-574	153	12	–	–	PUNCT
ejpam-574	153	13	a	a	X
ejpam-574	153	14	)	)	PUNCT
ejpam-574	153	15	=	=	SYM
ejpam-574	153	16	cl(a)–a	cl(a)–a	PROPN
ejpam-574	153	17	.	.	PROPN
ejpam-574	153	18	by	by	ADP
ejpam-574	153	19	hypothesis	hypothesis	NOUN
ejpam-574	153	20	,	,	PUNCT
ejpam-574	153	21	(	(	PUNCT
ejpam-574	153	22	cl(a)∩(x	cl(a)∩(x	NOUN
ejpam-574	153	23	–	–	PUNCT
ejpam-574	153	24	g	g	NOUN
ejpam-574	153	25	)	)	PUNCT
ejpam-574	153	26	)	)	PUNCT
ejpam-574	153	27	–	–	PUNCT
ejpam-574	154	1	u	u	PROPN
ejpam-574	154	2	⊂	⊂	PROPN
ejpam-574	154	3	int(cl(a)–a	int(cl(a)–a	PROPN
ejpam-574	154	4	)	)	PUNCT
ejpam-574	154	5	=	=	SYM
ejpam-574	154	6	∅	∅	NOUN
ejpam-574	154	7	,	,	PUNCT
ejpam-574	154	8	for	for	ADP
ejpam-574	154	9	some	some	DET
ejpam-574	154	10	u	u	PROPN
ejpam-574	154	11	∈	∈	PROPN
ejpam-574	154	12	i.	i.	NOUN
ejpam-574	154	13	this	this	PRON
ejpam-574	154	14	implies	imply	VERB
ejpam-574	154	15	that	that	SCONJ
ejpam-574	154	16	cl(a	cl(a	VERB
ejpam-574	154	17	)	)	PUNCT
ejpam-574	154	18	∩	∩	NOUN
ejpam-574	154	19	(	(	PUNCT
ejpam-574	154	20	x	x	X
ejpam-574	154	21	–	–	PUNCT
ejpam-574	154	22	g	g	NOUN
ejpam-574	154	23	)	)	PUNCT
ejpam-574	154	24	⊂	⊂	PROPN
ejpam-574	155	1	u	u	X
ejpam-574	155	2	∈	∈	PROPN
ejpam-574	156	1	i	i	PRON
ejpam-574	156	2	and	and	CCONJ
ejpam-574	156	3	hence	hence	ADV
ejpam-574	156	4	cl(a)–g	cl(a)–g	PROPN
ejpam-574	156	5	∈i	∈i	NUM
ejpam-574	156	6	.	.	PUNCT
ejpam-574	157	1	thus	thus	ADV
ejpam-574	157	2	,	,	PUNCT
ejpam-574	157	3	a	a	PRON
ejpam-574	157	4	is	be	AUX
ejpam-574	157	5	ig	ig	PRON
ejpam-574	157	6	-	-	ADJ
ejpam-574	157	7	closed	closed	ADJ
ejpam-574	157	8	.	.	PUNCT
ejpam-574	158	1	theorem	theorem	NOUN
ejpam-574	158	2	11	11	NUM
ejpam-574	158	3	.	.	PUNCT
ejpam-574	159	1	let	let	VERB
ejpam-574	159	2	f	f	X
ejpam-574	159	3	:	:	PUNCT
ejpam-574	159	4	(	(	PUNCT
ejpam-574	159	5	x	x	X
ejpam-574	159	6	,	,	PUNCT
ejpam-574	159	7	τ)→	τ)→	PROPN
ejpam-574	159	8	(	(	PUNCT
ejpam-574	159	9	y	y	PROPN
ejpam-574	159	10	,	,	PUNCT
ejpam-574	159	11	σ	σ	PROPN
ejpam-574	159	12	)	)	PUNCT
ejpam-574	159	13	be	be	AUX
ejpam-574	159	14	continuous	continuous	ADJ
ejpam-574	159	15	and	and	CCONJ
ejpam-574	159	16	closed	closed	ADJ
ejpam-574	159	17	.	.	PUNCT
ejpam-574	160	1	if	if	SCONJ
ejpam-574	160	2	a	a	DET
ejpam-574	160	3	⊂	⊂	X
ejpam-574	160	4	x	x	X
ejpam-574	160	5	is	be	AUX
ejpam-574	160	6	ig	ig	PRON
ejpam-574	160	7	-	-	VERB
ejpam-574	160	8	closed	closed	ADJ
ejpam-574	160	9	in	in	ADP
ejpam-574	160	10	x	x	NOUN
ejpam-574	160	11	,	,	PUNCT
ejpam-574	160	12	then	then	ADV
ejpam-574	160	13	f(a	f(a	PROPN
ejpam-574	160	14	)	)	PUNCT
ejpam-574	160	15	is	be	AUX
ejpam-574	160	16	f(i)-g	f(i)-g	ADV
ejpam-574	160	17	-	-	PUNCT
ejpam-574	160	18	closed	closed	ADJ
ejpam-574	160	19	in	in	ADP
ejpam-574	160	20	(	(	PUNCT
ejpam-574	160	21	y	y	PROPN
ejpam-574	160	22	,	,	PUNCT
ejpam-574	160	23	σ	σ	PROPN
ejpam-574	160	24	)	)	PUNCT
ejpam-574	160	25	,	,	PUNCT
ejpam-574	160	26	where	where	SCONJ
ejpam-574	160	27	f(i	f(i	NUM
ejpam-574	160	28	)	)	PUNCT
ejpam-574	161	1	=	=	PRON
ejpam-574	161	2	{	{	PUNCT
ejpam-574	161	3	f(u	f(u	PROPN
ejpam-574	161	4	):	):	PUNCT
ejpam-574	161	5	u	u	PROPN
ejpam-574	161	6	∈	∈	PROPN
ejpam-574	161	7	i	i	X
ejpam-574	161	8	}	}	PUNCT
ejpam-574	161	9	.	.	PUNCT
ejpam-574	162	1	proof	proof	NOUN
ejpam-574	162	2	.	.	PUNCT
ejpam-574	163	1	suppose	suppose	VERB
ejpam-574	163	2	a	a	DET
ejpam-574	163	3	⊂	⊂	PROPN
ejpam-574	163	4	x	x	X
ejpam-574	163	5	and	and	CCONJ
ejpam-574	163	6	a	a	PRON
ejpam-574	163	7	is	be	AUX
ejpam-574	163	8	ig	ig	PRON
ejpam-574	163	9	-	-	ADJ
ejpam-574	163	10	closed	closed	ADJ
ejpam-574	163	11	.	.	PUNCT
ejpam-574	164	1	suppose	suppose	VERB
ejpam-574	164	2	f(a	f(a	NOUN
ejpam-574	164	3	)	)	PUNCT
ejpam-574	165	1	⊂	⊂	PROPN
ejpam-574	165	2	g	g	PROPN
ejpam-574	165	3	and	and	CCONJ
ejpam-574	165	4	g	g	PROPN
ejpam-574	165	5	is	be	AUX
ejpam-574	165	6	open	open	ADJ
ejpam-574	165	7	.	.	PUNCT
ejpam-574	166	1	then	then	ADV
ejpam-574	166	2	a	a	DET
ejpam-574	166	3	⊂	⊂	PROPN
ejpam-574	166	4	f−1(g	f−1(g	PROPN
ejpam-574	166	5	)	)	PUNCT
ejpam-574	166	6	.	.	PUNCT
ejpam-574	167	1	by	by	ADP
ejpam-574	167	2	definition	definition	NOUN
ejpam-574	167	3	,	,	PUNCT
ejpam-574	167	4	cl(a	cl(a	NUM
ejpam-574	167	5	)	)	PUNCT
ejpam-574	167	6	–	–	PUNCT
ejpam-574	167	7	f−1(g	f−1(g	PROPN
ejpam-574	167	8	)	)	PUNCT
ejpam-574	167	9	∈i	∈i	NOUN
ejpam-574	167	10	and	and	CCONJ
ejpam-574	167	11	hence	hence	ADV
ejpam-574	167	12	f(cl(a	f(cl(a	ADJ
ejpam-574	167	13	)	)	PUNCT
ejpam-574	167	14	)	)	PUNCT
ejpam-574	167	15	–	–	PUNCT
ejpam-574	167	16	g	g	PROPN
ejpam-574	167	17	∈	∈	PROPN
ejpam-574	167	18	f(i	f(i	PROPN
ejpam-574	167	19	)	)	PUNCT
ejpam-574	167	20	.	.	PUNCT
ejpam-574	168	1	since	since	SCONJ
ejpam-574	168	2	f	f	PROPN
ejpam-574	168	3	is	be	AUX
ejpam-574	168	4	closed	closed	ADJ
ejpam-574	168	5	,	,	PUNCT
ejpam-574	168	6	cl(f(a	cl(f(a	NOUN
ejpam-574	168	7	)	)	PUNCT
ejpam-574	168	8	)	)	PUNCT
ejpam-574	169	1	⊂	⊂	PROPN
ejpam-574	169	2	cl(f(cl(a	cl(f(cl(a	PROPN
ejpam-574	169	3	)	)	PUNCT
ejpam-574	169	4	)	)	PUNCT
ejpam-574	169	5	)	)	PUNCT
ejpam-574	170	1	=	=	PUNCT
ejpam-574	170	2	f(cl(a	f(cl(a	ADJ
ejpam-574	170	3	)	)	PUNCT
ejpam-574	170	4	)	)	PUNCT
ejpam-574	170	5	.	.	PUNCT
ejpam-574	171	1	then	then	ADV
ejpam-574	171	2	cl(f(a	cl(f(a	NOUN
ejpam-574	171	3	)	)	PUNCT
ejpam-574	171	4	)	)	PUNCT
ejpam-574	171	5	–	–	PUNCT
ejpam-574	171	6	g	g	PROPN
ejpam-574	171	7	⊂	⊂	PROPN
ejpam-574	171	8	f(cl(a	f(cl(a	PROPN
ejpam-574	171	9	)	)	PUNCT
ejpam-574	171	10	)	)	PUNCT
ejpam-574	171	11	–	–	PUNCT
ejpam-574	171	12	g	g	PROPN
ejpam-574	171	13	∈	∈	PROPN
ejpam-574	171	14	f(i	f(i	PROPN
ejpam-574	171	15	)	)	PUNCT
ejpam-574	171	16	and	and	CCONJ
ejpam-574	171	17	hence	hence	ADV
ejpam-574	171	18	f(a	f(a	NOUN
ejpam-574	171	19	)	)	PUNCT
ejpam-574	171	20	is	be	AUX
ejpam-574	171	21	f(i)-g	f(i)-g	ADV
ejpam-574	171	22	-	-	PUNCT
ejpam-574	171	23	closed	closed	ADJ
ejpam-574	171	24	.	.	PUNCT
ejpam-574	172	1	references	reference	NOUN
ejpam-574	172	2	[	[	X
ejpam-574	172	3	1	1	X
ejpam-574	172	4	]	]	PUNCT
ejpam-574	172	5	t.	t.	PROPN
ejpam-574	172	6	r.	r.	PROPN
ejpam-574	172	7	hamlett	hamlett	PROPN
ejpam-574	172	8	and	and	CCONJ
ejpam-574	172	9	d.	d.	PROPN
ejpam-574	172	10	jankovic	jankovic	PROPN
ejpam-574	172	11	,	,	PUNCT
ejpam-574	172	12	compactness	compactness	NOUN
ejpam-574	172	13	with	with	ADP
ejpam-574	172	14	respect	respect	NOUN
ejpam-574	172	15	to	to	ADP
ejpam-574	172	16	an	an	DET
ejpam-574	172	17	ideal	ideal	ADJ
ejpam-574	172	18	,	,	PUNCT
ejpam-574	172	19	boll	boll	NOUN
ejpam-574	172	20	.	.	PUNCT
ejpam-574	173	1	un	un	PROPN
ejpam-574	173	2	.	.	PROPN
ejpam-574	173	3	mat	mat	PROPN
ejpam-574	173	4	.	.	PUNCT
ejpam-574	173	5	ita	ita	PROPN
ejpam-574	173	6	.	.	PROPN
ejpam-574	173	7	,	,	PUNCT
ejpam-574	173	8	(	(	PUNCT
ejpam-574	173	9	7	7	NUM
ejpam-574	173	10	)	)	PUNCT
ejpam-574	173	11	,	,	PUNCT
ejpam-574	173	12	4	4	NUM
ejpam-574	173	13	-	-	SYM
ejpam-574	173	14	b	b	NOUN
ejpam-574	173	15	,	,	PUNCT
ejpam-574	173	16	849	849	NUM
ejpam-574	173	17	-	-	SYM
ejpam-574	173	18	861	861	NUM
ejpam-574	173	19	.	.	NUM
ejpam-574	173	20	1990	1990	NUM
ejpam-574	173	21	.	.	PUNCT
ejpam-574	174	1	[	[	X
ejpam-574	174	2	2	2	X
ejpam-574	174	3	]	]	PUNCT
ejpam-574	174	4	t.	t.	PROPN
ejpam-574	174	5	r.	r.	PROPN
ejpam-574	174	6	hamlett	hamlett	PROPN
ejpam-574	174	7	and	and	CCONJ
ejpam-574	174	8	d.	d.	PROPN
ejpam-574	174	9	jankovic	jankovic	PROPN
ejpam-574	174	10	,	,	PUNCT
ejpam-574	174	11	ideals	ideal	NOUN
ejpam-574	174	12	in	in	ADP
ejpam-574	174	13	topological	topological	ADJ
ejpam-574	174	14	spaces	space	NOUN
ejpam-574	174	15	and	and	CCONJ
ejpam-574	174	16	the	the	DET
ejpam-574	174	17	set	set	NOUN
ejpam-574	174	18	operator	operator	NOUN
ejpam-574	174	19	,	,	PUNCT
ejpam-574	174	20	boll	boll	NOUN
ejpam-574	174	21	.	.	PUNCT
ejpam-574	175	1	un	un	PROPN
ejpam-574	175	2	.	.	PROPN
ejpam-574	175	3	mat	mat	PROPN
ejpam-574	175	4	.	.	PUNCT
ejpam-574	176	1	ita	ita	PROPN
ejpam-574	176	2	.	.	PROPN
ejpam-574	176	3	,	,	PUNCT
ejpam-574	176	4	7	7	NUM
ejpam-574	176	5	,	,	PUNCT
ejpam-574	176	6	863	863	NUM
ejpam-574	176	7	-	-	SYM
ejpam-574	176	8	874	874	NUM
ejpam-574	176	9	.	.	PUNCT
ejpam-574	177	1	1990	1990	NUM
ejpam-574	177	2	.	.	PUNCT
ejpam-574	178	1	[	[	X
ejpam-574	178	2	3	3	X
ejpam-574	178	3	]	]	PUNCT
ejpam-574	178	4	t.	t.	PROPN
ejpam-574	178	5	r.	r.	PROPN
ejpam-574	178	6	hamlett	hamlett	PROPN
ejpam-574	178	7	and	and	CCONJ
ejpam-574	178	8	d.	d.	PROPN
ejpam-574	178	9	jankovic	jankovic	PROPN
ejpam-574	178	10	,	,	PUNCT
ejpam-574	178	11	ideals	ideal	NOUN
ejpam-574	178	12	in	in	ADP
ejpam-574	178	13	general	general	ADJ
ejpam-574	178	14	topology	topology	NOUN
ejpam-574	178	15	and	and	CCONJ
ejpam-574	178	16	applications	application	NOUN
ejpam-574	178	17	(	(	PUNCT
ejpam-574	178	18	midletown	midletown	ADJ
ejpam-574	178	19	,	,	PUNCT
ejpam-574	178	20	ct	ct	PROPN
ejpam-574	178	21	,	,	PUNCT
ejpam-574	178	22	1988	1988	NUM
ejpam-574	178	23	)	)	PUNCT
ejpam-574	178	24	,	,	PUNCT
ejpam-574	178	25	115	115	NUM
ejpam-574	178	26	-	-	SYM
ejpam-574	178	27	125	125	NUM
ejpam-574	178	28	,	,	PUNCT
ejpam-574	178	29	lecture	lecture	NOUN
ejpam-574	178	30	notes	note	NOUN
ejpam-574	178	31	in	in	ADP
ejpam-574	178	32	pure	pure	ADJ
ejpam-574	178	33	and	and	CCONJ
ejpam-574	178	34	appl	appl	NOUN
ejpam-574	178	35	.	.	PROPN
ejpam-574	178	36	math	math	PROPN
ejpam-574	178	37	.	.	PUNCT
ejpam-574	179	1	dekker	dekker	PROPN
ejpam-574	179	2	,	,	PUNCT
ejpam-574	179	3	new	new	PROPN
ejpam-574	179	4	york	york	PROPN
ejpam-574	179	5	,	,	PUNCT
ejpam-574	179	6	1990	1990	NUM
ejpam-574	179	7	.	.	PUNCT
ejpam-574	180	1	references	reference	NOUN
ejpam-574	180	2	151	151	NUM
ejpam-574	181	1	[	[	SYM
ejpam-574	181	2	4	4	NUM
ejpam-574	181	3	]	]	PUNCT
ejpam-574	181	4	t.	t.	PROPN
ejpam-574	181	5	r.	r.	PROPN
ejpam-574	181	6	hamlett	hamlett	PROPN
ejpam-574	181	7	and	and	CCONJ
ejpam-574	181	8	d.	d.	PROPN
ejpam-574	181	9	jankovic	jankovic	PROPN
ejpam-574	181	10	,	,	PUNCT
ejpam-574	181	11	compatible	compatible	ADJ
ejpam-574	181	12	extensions	extension	NOUN
ejpam-574	181	13	of	of	ADP
ejpam-574	181	14	ideals	ideal	NOUN
ejpam-574	181	15	,	,	PUNCT
ejpam-574	181	16	boll	boll	NOUN
ejpam-574	181	17	.	.	PUNCT
ejpam-574	182	1	un	un	PROPN
ejpam-574	182	2	.	.	PROPN
ejpam-574	182	3	mat	mat	PROPN
ejpam-574	182	4	.	.	PUNCT
ejpam-574	183	1	ita	ita	PROPN
ejpam-574	183	2	.	.	PROPN
ejpam-574	183	3	,	,	PUNCT
ejpam-574	183	4	7	7	NUM
ejpam-574	183	5	,	,	PUNCT
ejpam-574	183	6	453	453	NUM
ejpam-574	183	7	-	-	SYM
ejpam-574	183	8	465	465	NUM
ejpam-574	183	9	.	.	PUNCT
ejpam-574	183	10	1992	1992	NUM
ejpam-574	183	11	.	.	PUNCT
ejpam-574	184	1	[	[	X
ejpam-574	184	2	5	5	X
ejpam-574	184	3	]	]	PUNCT
ejpam-574	184	4	d.	d.	PROPN
ejpam-574	184	5	jankovic	jankovic	PROPN
ejpam-574	184	6	and	and	CCONJ
ejpam-574	184	7	t.	t.	PROPN
ejpam-574	184	8	r.	r.	PROPN
ejpam-574	184	9	hamlett	hamlett	PROPN
ejpam-574	184	10	,	,	PUNCT
ejpam-574	184	11	new	new	ADJ
ejpam-574	184	12	topologies	topology	NOUN
ejpam-574	184	13	from	from	ADP
ejpam-574	184	14	old	old	ADJ
ejpam-574	184	15	via	via	ADP
ejpam-574	184	16	ideals	ideal	NOUN
ejpam-574	184	17	,	,	PUNCT
ejpam-574	184	18	amer	amer	PROPN
ejpam-574	184	19	.	.	PROPN
ejpam-574	184	20	math	math	PROPN
ejpam-574	184	21	.	.	PUNCT
ejpam-574	185	1	month	month	NOUN
ejpam-574	185	2	.	.	PUNCT
ejpam-574	186	1	,	,	PUNCT
ejpam-574	186	2	97	97	NUM
ejpam-574	186	3	,	,	PUNCT
ejpam-574	186	4	295	295	NUM
ejpam-574	186	5	-	-	SYM
ejpam-574	186	6	310	310	NUM
ejpam-574	186	7	.	.	NOUN
ejpam-574	187	1	1990	1990	NUM
ejpam-574	187	2	.	.	PUNCT
ejpam-574	188	1	[	[	X
ejpam-574	188	2	6	6	NUM
ejpam-574	188	3	]	]	PUNCT
ejpam-574	188	4	k.	k.	PROPN
ejpam-574	188	5	kuratowski	kuratowski	PROPN
ejpam-574	188	6	,	,	PUNCT
ejpam-574	188	7	topologies	topology	NOUN
ejpam-574	188	8	i	i	PRON
ejpam-574	188	9	,	,	PUNCT
ejpam-574	188	10	warszawa	warszawa	PROPN
ejpam-574	188	11	,	,	PUNCT
ejpam-574	188	12	1933	1933	NUM
ejpam-574	188	13	.	.	PUNCT
ejpam-574	189	1	[	[	X
ejpam-574	189	2	7	7	X
ejpam-574	189	3	]	]	X
ejpam-574	189	4	n.	n.	PROPN
ejpam-574	189	5	levine	levine	PROPN
ejpam-574	189	6	,	,	PUNCT
ejpam-574	189	7	generalized	generalize	VERB
ejpam-574	189	8	closed	closed	ADJ
ejpam-574	189	9	sets	set	NOUN
ejpam-574	189	10	in	in	ADP
ejpam-574	189	11	topology	topology	NOUN
ejpam-574	189	12	,	,	PUNCT
ejpam-574	189	13	rend	rend	VERB
ejpam-574	189	14	.	.	PUNCT
ejpam-574	190	1	circ	circ	PROPN
ejpam-574	190	2	.	.	PUNCT
ejpam-574	191	1	mat	mat	PROPN
ejpam-574	191	2	.	.	PUNCT
ejpam-574	191	3	palermo	palermo	PROPN
ejpam-574	191	4	,	,	PUNCT
ejpam-574	191	5	19(2	19(2	NUM
ejpam-574	191	6	)	)	PUNCT
ejpam-574	191	7	,	,	PUNCT
ejpam-574	191	8	89	89	NUM
ejpam-574	191	9	-	-	SYM
ejpam-574	191	10	96	96	NUM
ejpam-574	191	11	.	.	PUNCT
ejpam-574	191	12	1970	1970	NUM
ejpam-574	191	13	.	.	PUNCT
ejpam-574	192	1	[	[	X
ejpam-574	192	2	8	8	NUM
ejpam-574	192	3	]	]	X
ejpam-574	192	4	r.	r.	PROPN
ejpam-574	192	5	l.	l.	PROPN
ejpam-574	192	6	newcomb	newcomb	PROPN
ejpam-574	192	7	,	,	PUNCT
ejpam-574	192	8	topologies	topology	NOUN
ejpam-574	192	9	which	which	PRON
ejpam-574	192	10	are	be	AUX
ejpam-574	192	11	compact	compact	ADJ
ejpam-574	192	12	modulo	modulo	NOUN
ejpam-574	192	13	an	an	DET
ejpam-574	192	14	ideal	ideal	NOUN
ejpam-574	192	15	,	,	PUNCT
ejpam-574	192	16	ph.d	ph.d	PROPN
ejpam-574	192	17	.	.	PUNCT
ejpam-574	192	18	dissertation	dissertation	NOUN
ejpam-574	192	19	,	,	PUNCT
ejpam-574	192	20	univ	univ	PROPN
ejpam-574	192	21	.	.	PUNCT
ejpam-574	192	22	cal	cal	PROPN
ejpam-574	192	23	.	.	PUNCT
ejpam-574	193	1	at	at	ADP
ejpam-574	193	2	santa	santa	PROPN
ejpam-574	193	3	barbara	barbara	PROPN
ejpam-574	193	4	,	,	PUNCT
ejpam-574	193	5	1967	1967	NUM
ejpam-574	193	6	.	.	PUNCT
ejpam-574	194	1	[	[	X
ejpam-574	194	2	9	9	NUM
ejpam-574	194	3	]	]	X
ejpam-574	194	4	d.	d.	PROPN
ejpam-574	194	5	v.	v.	PROPN
ejpam-574	194	6	rancin	rancin	PROPN
ejpam-574	194	7	,	,	PUNCT
ejpam-574	194	8	compactness	compactness	NOUN
ejpam-574	194	9	modulo	modulo	VERB
ejpam-574	194	10	an	an	DET
ejpam-574	194	11	ideal	ideal	ADJ
ejpam-574	194	12	,	,	PUNCT
ejpam-574	194	13	soviet	soviet	ADJ
ejpam-574	194	14	math	math	NOUN
ejpam-574	194	15	.	.	PUNCT
ejpam-574	195	1	dokl	dokl	NOUN
ejpam-574	195	2	.	.	PUNCT
ejpam-574	196	1	,	,	PUNCT
ejpam-574	196	2	13	13	NUM
ejpam-574	196	3	,	,	PUNCT
ejpam-574	196	4	193	193	NUM
ejpam-574	196	5	-	-	SYM
ejpam-574	196	6	197	197	NUM
ejpam-574	196	7	.	.	PUNCT
ejpam-574	197	1	1972	1972	NUM
ejpam-574	197	2	.	.	PUNCT
ejpam-574	198	1	[	[	X
ejpam-574	198	2	10	10	NUM
ejpam-574	198	3	]	]	X
ejpam-574	198	4	p.	p.	PROPN
ejpam-574	198	5	samuels	samuels	PROPN
ejpam-574	198	6	,	,	PUNCT
ejpam-574	198	7	a	a	DET
ejpam-574	198	8	topology	topology	NOUN
ejpam-574	198	9	from	from	ADP
ejpam-574	198	10	a	a	DET
ejpam-574	198	11	given	give	VERB
ejpam-574	198	12	topology	topology	NOUN
ejpam-574	198	13	and	and	CCONJ
ejpam-574	198	14	ideal	ideal	ADJ
ejpam-574	198	15	,	,	PUNCT
ejpam-574	198	16	j.	j.	PROPN
ejpam-574	198	17	london	london	PROPN
ejpam-574	198	18	math	math	PROPN
ejpam-574	198	19	.	.	PUNCT
ejpam-574	199	1	soc	soc	PROPN
ejpam-574	199	2	.	.	PUNCT
ejpam-574	200	1	(	(	PUNCT
ejpam-574	200	2	2)(10	2)(10	NUM
ejpam-574	200	3	)	)	PUNCT
ejpam-574	200	4	,	,	PUNCT
ejpam-574	200	5	409	409	NUM
ejpam-574	200	6	-	-	SYM
ejpam-574	200	7	416	416	NUM
ejpam-574	200	8	.	.	PUNCT
ejpam-574	200	9	1975	1975	NUM